# Difference between revisions of "Math 302: Math for Engineering 1"

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(Added learning outcomes from course materials) |
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=== Prerequisite === | === Prerequisite === | ||

− | [[Math 113]] and passing grade on required preparatory exam taken during first week of class. (Practice exams available on class | + | [[Math 113]] and passing grade on required preparatory exam taken during first week of class. (Practice exams available on class website.) |

=== Description === | === Description === | ||

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== Desired Learning Outcomes == | == Desired Learning Outcomes == | ||

+ | This course is designed to give students from the College of Engineering and Technology the mathematics background necessary to succeed in their chosen field. | ||

=== Prerequisites === | === Prerequisites === | ||

+ | |||

+ | Students are expected to have completed [[Math 113]]. | ||

=== Minimal learning outcomes === | === Minimal learning outcomes === | ||

<div style="-moz-column-count:2; column-count:2;"> | <div style="-moz-column-count:2; column-count:2;"> | ||

+ | |||

+ | # Rectangular Space Coordinates; Vectors in Three-Dimensional Space | ||

+ | #* Define the following: | ||

+ | #** Cartesian coordinates of a point | ||

+ | #** sphere | ||

+ | #** symmetry about a point, a line, and a plane | ||

+ | #** vector | ||

+ | #** components of a vector | ||

+ | #** vector addition | ||

+ | #** scalar multiplication | ||

+ | #** zero vector | ||

+ | #** vector subtraction | ||

+ | #** vector norm (magnitude, length) | ||

+ | #** unit vector | ||

+ | #** coordinate unit vectors i, j, k | ||

+ | #** linear combination of unit vectors | ||

+ | #* Plot points in three-dimensional space. | ||

+ | #* Calculate the distance between two points in two-dimensional space and 3-dimensional space | ||

+ | #* Write the equation of a sphere centered about a given point with a given radius. Determine the center and radius of a sphere, given its equation. | ||

+ | #* Write the component equations of a line that passes through two given points. | ||

+ | #* Write the component equations of a line segment with given endpoints. | ||

+ | #* Find the midpoint of a given line segment. | ||

+ | #* Find the points of symmetry about a point, line, or plane. | ||

+ | #* Represent a vector by each of the following: | ||

+ | #** components | ||

+ | #** a linear combination of coordinate unit vectors | ||

+ | #* Carry out the vector operations: | ||

+ | #** addition | ||

+ | #** scalar multiplication | ||

+ | #** subtraction | ||

+ | #* Represent the operations of vector addition, scalar multiplication and norm geometrically. | ||

+ | #* Find the norm (magnitude, length) of a vector. Determine whether two vectors are parallel. | ||

+ | #* Recall, apply and verify the basic properties of vector addition, scalar multiplication and norm. | ||

+ | #* Model and solve application problems using vectors. | ||

+ | # The Dot Product | ||

+ | #* Define the following: | ||

+ | #** dot product. | ||

+ | #** perpendicular vectors. | ||

+ | #** unit vector in the direction of a vector a, denoted u_a. | ||

+ | #** the projection of a on b, denoted proj_b a. | ||

+ | #** the b-component of a, denoted comp_b a. | ||

+ | #** the direction cosines of a vector. | ||

+ | #** the direction angles of a vector. | ||

+ | #** the Schwarz Inequality. | ||

+ | #** the work done by a constant force on an object. | ||

+ | #** the dot product test for perpendicular vectors. | ||

+ | #** the dot product test for parallel vectors. | ||

+ | #** geometric interpretation of the dot product | ||

+ | #* Evaluate a dot product from the coordinate formula or the angle formula. | ||

+ | #* Interpret the dot product geometrically. | ||

+ | #* Evaluate the following using the dot product: | ||

+ | #** the length of a vector. | ||

+ | #** the angle between two vectors. | ||

+ | #** u_a, the unit vector in the direction of a vector a. | ||

+ | #** proj_b a, the projection of a on b. | ||

+ | #** comp_b a, the b-component of a. | ||

+ | #** the direction cosines of a vector. | ||

+ | #** the direction angles of a vector. | ||

+ | #** the work done by a constant force on an object. | ||

+ | #* Prove and verify the Schwarz Inequality. | ||

+ | #* Prove and apply the dot product tests for perpendicular and parallel vectors. | ||

+ | #* Recall and apply the properties of the dot product. | ||

+ | #* Prove identities involving the dot product. | ||

+ | #* Solve application problems involving the dot product. | ||

+ | #* Extend the vector operations and related identities for addition, scalar multiplication, and dot product to higher dimensions. | ||

+ | # The Cross Product | ||

+ | #* Define the following: | ||

+ | #** the cross product of two vectors | ||

+ | #** scalar triple product | ||

+ | #* Evaluate a cross product from the the coordinate formula or angle formula. | ||

+ | #* Interpret the cross product geometrically. | ||

+ | #* Evaluate the following using the cross product: | ||

+ | #** a vector perpendicular to two given vectors. | ||

+ | #** the area of a parallelogram. | ||

+ | #** the area or a triangle. | ||

+ | #** moment of force or moment of torque. | ||

+ | #* Evaluate scalar triple products. | ||

+ | #* Use the scalar triple product to determine the following: | ||

+ | #** volume of a parallelepiped. | ||

+ | #** whether or not three vectors are coplanar. | ||

+ | #* Recall and apply the properties of the cross product and scalar triple product. | ||

+ | #* Prove identities involving the cross product and the scalar triple product. | ||

+ | #* Solve application problems involving the cross product and scalar triple product. | ||

+ | # Lines | ||

+ | #*Define the following: | ||

+ | #** direction vector for a line | ||

+ | #** vector equation of a line | ||

+ | #** scalar parametric equations of a line | ||

+ | #** Cartesian equations or symmetric form of a line | ||

+ | #* Represent a line in 3-space by: | ||

+ | #** a vector equation | ||

+ | #** scalar parametric equations | ||

+ | #** Cartesian equations | ||

+ | #* Find the equation(s) representing a line given information about | ||

+ | #** a point of the line and the direction of the line or | ||

+ | #** two points contained in the line. | ||

+ | #** a point and a parallel line. | ||

+ | #** a point and perpendicular to a plane. | ||

+ | #** two planes intersecting in the line. | ||

+ | #* Find the distance from a point to a line. | ||

+ | #* Solve application problems involving lines. | ||

+ | # Planes | ||

+ | #* Define the following: | ||

+ | #** normal vector to a plane | ||

+ | #** cartesian equation of a plane | ||

+ | #** parametric equation of a plane | ||

+ | #* Find the equation of a plane in 3-space given a point and a normal vector, three points, or a geometric description of the plane. | ||

+ | #* Determine a normal vector and the intercepts of a given plane. | ||

+ | #* Represent a plane by parametric equations. | ||

+ | #* Find the distance from a point to a plane. | ||

+ | #* Find the angle between a line and a plane. | ||

+ | #* Determine a point of intersection between a line and a surface. | ||

+ | #* Sketch planes given their equations. | ||

+ | #* Solve application problems involving planes. | ||

+ | # Systems of Linear Equations | ||

+ | #* Define the following: | ||

+ | #** linear system of m equations in n unknowns | ||

+ | #** consistent and inconsistent | ||

+ | #** solution set | ||

+ | #** coefficient matrix | ||

+ | #** elementary row operations | ||

+ | #* Identify linear systems. | ||

+ | #* Represent a system of linear equations as an augmented matrix and vice versa. | ||

+ | #* Relate the following types of solution sets of a system of two or three variables to the intersections of lines in a plane or the intersection of planes in three space: | ||

+ | #** a unique solution. | ||

+ | #** infinitely many solutions. | ||

+ | #** no solution. | ||

+ | # Gaussian elimination | ||

+ | #* Define the following: | ||

+ | #** reduced row echelon form | ||

+ | #** leading variables or pivots | ||

+ | #** free variables | ||

+ | #** row echelon form | ||

+ | #** back substitution | ||

+ | #** Gaussian elimination | ||

+ | #** Gauss-Jordan elimination | ||

+ | #** homogeneous | ||

+ | #** trivial solution | ||

+ | #** nontrivial solutions | ||

+ | #* Identify matrices that are in row echelon form and reduced row echelon form. | ||

+ | #* Determine whether a linear system is consistent or inconsistent from its reduced row echelon form. If the system is consistent, write the solution. | ||

+ | #* Identify the lead variables and free variables of a system represented by an augmented matrix in reduced row echelon form. | ||

+ | #* Solve systems of linear equations using Gaussian elimination and back substitution. | ||

+ | #* Solve systems of linear equations using Gauss-Jordan elimination. | ||

+ | #* Model and solve application problems using linear systems. | ||

+ | # Matrices and Matrix Operations | ||

+ | #* Define the following: | ||

+ | #** vector, row vector, and column vector | ||

+ | #** equal matrices | ||

+ | #** scalar multiplication | ||

+ | #** sum of matrices | ||

+ | #** zero matrix | ||

+ | #** scalar product | ||

+ | #** linear combination | ||

+ | #** matrix multiplication | ||

+ | #** transpose | ||

+ | #** trace | ||

+ | #** identity matrix | ||

+ | #* Perform the operations of matrix addition, scalar multiplication, transposition, trace, and matrix multiplication. | ||

+ | #* Represent matrices in terms of double subscript notation. | ||

+ | # Inverses; Rules of Matrix Arithmetic | ||

+ | #* Define the following: | ||

+ | #** commutative property | ||

+ | #** singular | ||

+ | #** nonsingular or invertible | ||

+ | #** multiplicative inverse | ||

+ | #* Recall, demonstrate, and apply algebraic properties for matrices. | ||

+ | #* Recall that matrix multiplication is not commutative in general. Determine conditions under which matrices do commute. | ||

+ | #* Recall and prove properties and identities involving the transpose operator. | ||

+ | #* Recall and prove properties and identities involving matrix inverses. | ||

+ | #* Recall and prove properties and identities involving matrix powers. | ||

+ | #* Recall, demonstrate, and apply that the cancelation laws for scalar multiplication do not hold for matrix multiplication. | ||

+ | #* Recall and apply the formula for the inverse of 2x2 matrices. | ||

+ | # Elementary Matrices | ||

+ | #* Define the following: | ||

+ | #** elementary matrix | ||

+ | #** row equivalent matrices | ||

+ | #* Identify elementary matrices and find their inverses or show that their inverse does not exist. | ||

+ | #* Relate elementary matrices to row operations. | ||

+ | #* Factor matrices using elementary matrices. | ||

+ | #* Find the inverse of a matrix, if possible, using elementary matrices. | ||

+ | #* Prove theorems about matrix products and matrix inverses. | ||

+ | #* Solve a linear equation using matrix inverses. | ||

+ | # Further Results on Systems of Equations and Invertibility | ||

+ | #* Solve matrix equations using matrix algebra. | ||

+ | #* Recall and prove properties and identities involving matrix inverses. | ||

+ | #* Recall equivalent conditions for invertibility. | ||

+ | # Further Results on Systems of Equations and Invertibility | ||

+ | #* Define the following: | ||

+ | #** diagonal matrix | ||

+ | #** upper and lower triangular matrices | ||

+ | #** symmetric matrix | ||

+ | #** skew-symmetric matrix | ||

+ | #* Determine powers of diagonal matrices. | ||

+ | #* Recall and prove properties and identities involving the transpose operator. | ||

+ | #* Prove basic facts involving symmetric and skew-symmetric matrices. | ||

+ | # Determinants | ||

+ | #* Define the following: | ||

+ | #** minor | ||

+ | #** cofactor | ||

+ | #** cofactor expansion | ||

+ | #** determinant | ||

+ | #** adjoint | ||

+ | #** Cramer's Rule | ||

+ | #* Apply cofactor expansion to evaluate determinants of nxn matrices. | ||

+ | #* Recall and apply the properties of determinants to evaluate determinants. | ||

+ | #* Evaluate the adjoint of a matrix. | ||

+ | #* Determine whether or not a matrix has an inverse based on its determinant. | ||

+ | #* Evaluate the inverse of a matrix using the adjoint method. | ||

+ | #* Use Cramer's rule to solve a linear system. | ||

+ | # Properties of Determinants | ||

+ | #* Recall the effects that row operations have on the determinants of matrices. Relate to the determinants of elementary matrices. | ||

+ | #* Recall, apply and verify the properties of determinants to evaluate determinants, including: | ||

+ | #** det(AB) = det(A) det(B) | ||

+ | #** det(kA) = k^n det(A) | ||

+ | #** det(A^-1)= 1/det(A) | ||

+ | #** det(A^T) = det(A) | ||

+ | #** det(A) = 0 if and only if A is singular | ||

+ | #* Evaluate the determinant of a matrix using row operations. | ||

+ | #* Apply determinants to determine invertibility of matrix products. | ||

+ | # Linear Transformations: Definitions and Examples | ||

+ | #* Define the following: | ||

+ | #** linear transformation | ||

+ | #** image | ||

+ | #** range | ||

+ | #* Describe geometrically the effects of a linear operator. | ||

+ | #* Determine whether or not a given transformation is linear. | ||

+ | #* Prove theorems and solve application problems involving linear transformations. | ||

+ | # Matrix Representations of Linear Transformations | ||

+ | #* Define the following: | ||

+ | #** standard matrix representation | ||

+ | #** eigenvalues and eigenvectors | ||

+ | #* Determine the matrix that represents a given linear transformation of vectors given an algebraic description. | ||

+ | #* Determine the matrix that represents a given linear transformation of vectors given a geometric description. | ||

+ | #* Prove theorems and solve application problems involving linear transformations. | ||

+ | # Vector Spaces: Definitions and Examples | ||

+ | #* Define the following: | ||

+ | #** vector space | ||

+ | #** vector space axioms | ||

+ | #** vector space R^n | ||

+ | #** vector space R^(mxn) | ||

+ | #** vector space of real-valued functions | ||

+ | #** additional properties of vector spaces | ||

+ | #* Prove or disprove that a given set of vectors together with an addition and a scalar multiplication is a vector space. | ||

+ | #* Prove and verify properties of a vector space. | ||

+ | # Subspaces | ||

+ | #* Define the following: | ||

+ | #** subspace | ||

+ | #** closure under addition | ||

+ | #** closure under scalar multiplication | ||

+ | #** zero subspace | ||

+ | #** linear combination | ||

+ | #** span (or subspace spanned by a set of vectors) | ||

+ | #** spanning set | ||

+ | #* Prove or disprove that a set of vectors forms a subspace. | ||

+ | #* Prove or disprove a set of vectors is a spanning set for R^n. | ||

+ | #* Prove or disprove a given vector is in the span of a set of vectors. Determine the span of a set of vectors. | ||

+ | #* Prove theorems about vector spaces and spans. | ||

+ | # Linear Independence | ||

+ | #* Define the following: | ||

+ | #** linearly independent | ||

+ | #** linearly dependent | ||

+ | #** Wronskian | ||

+ | #* Determine whether a set of vectors is linearly dependent or linearly independent. | ||

+ | #* Geometrically describe the span of a set of vectors. For sets that are linearly dependent, determine a dependence relation. | ||

+ | #* Prove theorems about linear independence. | ||

+ | # Basis and Dimension | ||

+ | #* Define the following: | ||

+ | #** basis | ||

+ | #** dimension | ||

+ | #** finite and infinite dimensional | ||

+ | #** standard basis | ||

+ | #* Prove or disprove a set of vectors forms a basis. | ||

+ | #* Find a basis for a vector space. | ||

+ | #* Determine the dimension of a vector space. | ||

+ | #* Geometrically interpret the ideas of span, linear dependance, basis, and dimension. | ||

+ | # Row Space, Column Space, and Null Space | ||

+ | #* Define the following: | ||

+ | #** row space | ||

+ | #** column space | ||

+ | #** null space | ||

+ | #** particular solution | ||

+ | #** general solution | ||

+ | #* Express a product Ax as a linear combination of column vectors. | ||

+ | #* Find a basis for a the column space, the row space, and the null space of a matrix. | ||

+ | #* Find the basis for a span of vectors. | ||

+ | # Rank and Nullity | ||

+ | #* Define the following: | ||

+ | #** rank | ||

+ | #** nullity | ||

+ | #** The Consistency Theorem | ||

+ | #** equivalent statements of invertibility | ||

+ | #* Find the rank and nullity of a matrix. | ||

+ | #* Recall and prove identities involving rank and nullity | ||

+ | #* Recall and apply the Consistency Theorm | ||

+ | #* Recall and apply the equivalent statements of invertibility. | ||

+ | # Eigenvalues and Eigenvectors | ||

+ | #* Define the following: | ||

+ | #** eigenvalue or characteristic value | ||

+ | #** eigenvector or characteristic vector | ||

+ | #** characteristic polynomial or characteristic polynomial | ||

+ | #** equivalent statements of invertibility | ||

+ | #* Find the eigenvalues and eigenvectors of an nxn matrix. | ||

+ | #* Prove theorems and solve application problems involving eigenvalues and eigenvectors. | ||

+ | # Diagonalization | ||

+ | #* Define the following: | ||

+ | #** diagonalizable | ||

+ | #** algebraic multiplicity | ||

+ | #** geometric multiplicity | ||

+ | #* Determine whether or not a matrix is diagonalizable. | ||

+ | #* Find the diagonalization of a matrix, if possible. | ||

+ | #* Find powers of a matrix using the diagonalization of a matrix. | ||

+ | #* Prove theorems and solve application problems involving the diagonalization of matrices. | ||

+ | # Limit, Continuity, Vector Derivative; The Rules of Differentiation | ||

+ | #* Define the following: | ||

+ | #** scalar functions | ||

+ | #** vector functions | ||

+ | #** components of a vector function | ||

+ | #** plane curve or space curve | ||

+ | #** parametrization of a curve | ||

+ | #** limit of a vector function | ||

+ | #** a vector function continuous at a point | ||

+ | #** derivative of a vector function | ||

+ | #** a differentiable vector function | ||

+ | #** integral of a vector function | ||

+ | #* Graph a parametric curve. | ||

+ | #* Identify a curve given its parametrization. | ||

+ | #* Determine combinations of vector functions such as sums, vector products and scalar products. | ||

+ | #* Evaluate limits, derivatives, and integrals of vector functions. | ||

+ | #* Recall, derive and apply rules to combinations of vector functions for the following: | ||

+ | #** limits | ||

+ | #** differentiation | ||

+ | #** integration | ||

+ | #* Determine continuity of a vector-valued function. | ||

+ | #* Prove theorems involving limits and derivatives of vector-valued functions. | ||

+ | #* Solve application problems involving vector-valued functions. | ||

+ | # Curves; Vector Calculus in Mechanics | ||

+ | #* Define the following: | ||

+ | #** directed path | ||

+ | #** differentiable parameterized curve | ||

+ | #** tangent vector | ||

+ | #** tangent line | ||

+ | #** unit tangent vector | ||

+ | #** principal normal vector | ||

+ | #** normal line | ||

+ | #** osculation plane | ||

+ | #** force vector | ||

+ | #** momentum vector | ||

+ | #** angular momentum vector | ||

+ | #** torque | ||

+ | #* Find the tangent vector and tangent line to a curve at a given point. | ||

+ | #* Find the principle normal and normal line to a curve at a given point. | ||

+ | #* Determine the osculating plane for a space curve at a given point. | ||

+ | #* Reverse the direction of a curve. | ||

+ | #* Solve application problems involving curves. | ||

+ | #* Solve application problems involving force, momentum, angular momentum, and torque. | ||

+ | # Arc Length | ||

+ | #* Define the following: | ||

+ | #** arc length | ||

+ | #** arc length parametrization | ||

+ | #* Evaluate the arc length of a curve. | ||

+ | #* Determine whether a curve is arc length parameterized. | ||

+ | #* Find the arc length parametrization of a curve. | ||

+ | # Curvilinear Motion; Curvature | ||

+ | #* Define the following: | ||

+ | #** velocity vector function | ||

+ | #** speed | ||

+ | #** acceleration vector function | ||

+ | #** uniform circular motion | ||

+ | #** curvature | ||

+ | #** tangential component of acceleration | ||

+ | #** normal component of acceleration | ||

+ | #* Given the position vector function of a moving object, calculate the velocity vector function, speed, and acceleration vector function, and vice versa. | ||

+ | #* Calculate the curvature of a space curve. | ||

+ | #* Recall the formulas for the curvature of a parameterized planar curve or a planar curve that is the graph of a function. Apply these formulas to calculate the curvature of a planar curve. | ||

+ | #* Determine the tangential and normal components of acceleration for a given parameterized curve. | ||

+ | #* Solve application problems involving curvilinear motion and curvature. | ||

+ | # Functions of Several Variables; A Brief Catalogue of the Quadric Surfaces; Projections | ||

+ | #* Define the following: | ||

+ | #** real-valued function of several variables | ||

+ | #** domain | ||

+ | #** range | ||

+ | #** bounded functions | ||

+ | #** quadric surface | ||

+ | #** intercepts | ||

+ | #** traces | ||

+ | #** sections | ||

+ | #** center | ||

+ | #** symmetry | ||

+ | #** boundedness | ||

+ | #** cylinder | ||

+ | #** ellipsiod | ||

+ | #** elliptic cone | ||

+ | #** elliptic paraboloid | ||

+ | #** hyperboloid of one sheet | ||

+ | #** hyperboloid of two sheets | ||

+ | #** hyperbolic paraboloid | ||

+ | #** parabolic cylinder | ||

+ | #** elliptic cylinder | ||

+ | #** projection of a curve onto a coordinate plane | ||

+ | #* Describe the domain and range of a function of several variables. | ||

+ | #* Write a function of several variables given a description. | ||

+ | #* Identify standard quadratic surfaces given their functions or graphs. | ||

+ | #* Sketch the graph of a quadratic surface by sketching intercepts, traces, sections, centers, symmetry, boundedness. | ||

+ | #* Find the projection of a curve, that is the intersection of two surfaces, to a coordinate plane. | ||

+ | # Graphs; Level Curves and Level Surfaces | ||

+ | #* Define the following: | ||

+ | #** level curve | ||

+ | #** level surface | ||

+ | #* Describe the level sets of a function of several variables. | ||

+ | #* Graphically represent a function of two variables by level curves or a function of three variables by level surfaces. | ||

+ | #* Identify the characteristics of a function from its graph or from a graph of its level curves (or level surfaces). | ||

+ | #* Solve application problems involving level sets. functions. | ||

+ | # Partial Derivatives | ||

+ | #* Define the following: | ||

+ | #** partial derivative of a function of several variables | ||

+ | #** second partial derivative | ||

+ | #** mixed partial derivative | ||

+ | #* Interpret the definition of a partial derivative of a function of two variables graphically. | ||

+ | #* Evaluate the partial derivatives of a function of several variables. | ||

+ | #* Evaluate the higher order partial derivatives of a function of several variables. | ||

+ | #* Verify equations involving partial derivatives. | ||

+ | #* Apply partial derivatives to solve application problems. | ||

+ | # Open and Closed Sets; Limits and Continuity; Equity of Mixed Partials | ||

+ | #* Define the following: | ||

+ | #** neighborhood of a point | ||

+ | #** deleted neighborhood of a point | ||

+ | #** interior of a set | ||

+ | #** boundary of a set | ||

+ | #** open set | ||

+ | #** closed set | ||

+ | #** limit of a function of several variables at a point | ||

+ | #** continuity of a function of several variables at a point | ||

+ | #* Determine the boundary and interior of a set. | ||

+ | #* Determine whether a set is open, closed, neither, or both. | ||

+ | #* Evaluate the limit of a function of several variables or show that it does not exists. | ||

+ | #* Determine whether or not a function is continuous at a given point. | ||

+ | #* Recall and apply the conditions under which mixed partial derivatives are equal. | ||

+ | # Differentiability and Gradient | ||

+ | #* Define the following: | ||

+ | #** differentiable multivariable function | ||

+ | #** gradient of a multivariable function | ||

+ | #* Evaluate the gradient of a function. | ||

+ | #* Find a function with a given gradient. | ||

+ | # Gradient and Directional Derivative | ||

+ | #* Define the following: | ||

+ | #** directional derivative | ||

+ | #** isothermals | ||

+ | #* Recall and prove identities involving gradients. | ||

+ | #* Give a graphical interpretation of the gradient. | ||

+ | #* Evaluate the directional derivative of a function. | ||

+ | #* Give a graphical interpretation of directional derivative. | ||

+ | #* Recall, prove, and apply the theorem that states that a differential function f increases most rapidly in the direction of the gradient (the rate of change is then ||f(x)||) and it decreases most rapidly in the opposite direction (the rate of change is then -||f(x)||). | ||

+ | #* Find the path of a heat seeking or a heat repelling particle. | ||

+ | #* Solve application problems involving gradient and directional derivatives. | ||

+ | # The Mean-Value Theorem; The Chain Rule | ||

+ | #* Define the following: | ||

+ | #** the Mean Value Theorem for functions of several variables | ||

+ | #** normal line | ||

+ | #** chain rules for functions of several variables | ||

+ | #** implicit differentiation | ||

+ | #* Recall and apply the Mean Value Theorem for functions of several variables and its corollaries. | ||

+ | #* Apply an appropriate chain rule to evaluate a rate of change. | ||

+ | #* Apply implicit differentiation to evaluate rates of change. | ||

+ | #* Solve application problems involving chain rules and implicit differentiation. | ||

+ | # The Gradient as a Normal; Tangent Lines and Tangent Planes | ||

+ | #* Define the following: | ||

+ | #** normal vector | ||

+ | #** tangent vector | ||

+ | #** tangent line | ||

+ | #** tangent plane | ||

+ | #** normal line | ||

+ | #* Use gradients to find the normal vector and normal line to a smooth planar curve at a given point. | ||

+ | #* Use gradients to find the tangent vector and tangent line to a smooth planar curve at a given point. | ||

+ | #* Use gradients to find the normal vector to a smooth surface at a given point. | ||

+ | #* Use gradients to find the tangent plane to a smooth surface at a given point. | ||

+ | #* Use gradients to find the normal line to a smooth surface at a given point. | ||

+ | #* Solve application problems involving normals and tangents to curves and surfaces. | ||

+ | # Local Extreme Values | ||

+ | #* Define the following: | ||

+ | #** local minimum and local maximum | ||

+ | #** critical points | ||

+ | #** stationary points | ||

+ | #** saddle points | ||

+ | #** discriminant | ||

+ | #** Second Derivative Test | ||

+ | #* Find the critical points of a function of two variables. | ||

+ | #* Apply the Second-Partials Test to determine whether each critical point is a local minimum, a local maximum, or a saddle point. | ||

+ | #* Solve word problems involving local extreme values. | ||

+ | # Absolute Extreme Values | ||

+ | #* Define the following: | ||

+ | #** absolute minimum and absolute maximum | ||

+ | #** bounded subset of a plane or three-space | ||

+ | #** the Extreme Value Theorem | ||

+ | #* Determine absolute extreme values of a function defined on a closed and bounded set. | ||

+ | #* Apply the Extreme Value Theorem to justify the method for finding extreme values of functions defined on certain sets. | ||

+ | #* Solve word problems involving absolute extreme values. | ||

+ | # Maxima and Minima with Side Conditions | ||

+ | #* Define the following: | ||

+ | #** side conditions or constraints | ||

+ | #** method of Lagrange | ||

+ | #** Lagrange multipliers | ||

+ | #** cross-product equation of the Lagrange condition | ||

+ | #* Graphically interpret the method of Lagrange. | ||

+ | #* Determine the extreme values of a function subject to a side conditions by applying the method of Lagrange. | ||

+ | #* Apply the cross-product equation of the Lagrange condition to solve extreme value problems subject to side conditions. | ||

+ | #* Apply the method of Lagrange to solve word problems. | ||

+ | # Differentials; Reconstructing a Function from its Gradient | ||

+ | #* Define the following: | ||

+ | #** differential | ||

+ | #** general solution | ||

+ | #** particular solution | ||

+ | #** connected open set | ||

+ | #** open region | ||

+ | #** simple closed curve | ||

+ | #** simply connected open region | ||

+ | #** partial derivative gradient test | ||

+ | #* Determine the differential for a given function of several variables. | ||

+ | #* Determine whether or not a vector function is a gradient. | ||

+ | #* Given a vector function that is a gradient, find the functions with that gradient. | ||

+ | # Multiple-Sigma Notation; The Double Integral over a Rectangle R; The Evaluation of Double Integrals by Repeated Integrals | ||

+ | #* Define the following: | ||

+ | #** double sigma notation | ||

+ | #** triple sigma notation | ||

+ | #** upper sum | ||

+ | #** lower sum | ||

+ | #** double integral | ||

+ | #** integral formula for the volume of a solid bounded between a region Omega in the xy-plane and the graph of a non-negative function z = f(x,y) defined on Omega. | ||

+ | #** integral formula for the area of region in a plane | ||

+ | #** integral formula for the average of a function defined on a region Omega. | ||

+ | #** projection of a region onto a coordinate axis | ||

+ | #** Type I and Type II regions | ||

+ | #** reduction formulas for double integrals | ||

+ | #** the geometric interpretation of the reduction formulas for double integrals | ||

+ | #* Evaluate double and triple sums given their sigma notation. | ||

+ | #* Recall and apply summation identities. | ||

+ | #* Approximate the integral of a function by a lower sum and an upper sum. | ||

+ | #* Evaluate the integral of a function using the definition. | ||

+ | #* Evaluate double integrals over a rectangle using the reduction formulas. | ||

+ | #* Sketch planar regions and determine if they are Type I, Type II, or both. | ||

+ | #* Evaluate double integrals over Type I and Type II regions. | ||

+ | #* Change the order of integration of an integral. | ||

+ | #* Apply double integrals to calculate volumes, areas, and averages. | ||

+ | # The Double Integral as the Limit of Riemann Sums; Polar Coordinates | ||

+ | #* Define the following: | ||

+ | #** diameter of a set | ||

+ | #** Riemann sum | ||

+ | #** double integral as a limit of Riemann sums | ||

+ | #** polar coordinates (r; theta) | ||

+ | #** transformation formulas between Cartesian and polar coordinates | ||

+ | #** double integral conversion formula between Cartesian and polar coordinates | ||

+ | #* Represent a region in both Cartesian and polar coordinates. | ||

+ | #* Evaluate double integrals in terms of polar coordinates. | ||

+ | #* Evaluate areas and volumes using polar coordinates. | ||

+ | #* Convert a double integral in Cartesian coordinates to a double integral in polar coordinates and then evaluate. | ||

+ | # Further Applications of the Double Integral | ||

+ | #* Define the following: | ||

+ | #** integral formula for the mass of a plate | ||

+ | #** integral formulas for the center of mass of a plate | ||

+ | #** integral formulas for the centroid of a plate | ||

+ | #** integral formulas for the moment of an inertia of a plate | ||

+ | #** radius of gyration | ||

+ | #** the Parallel Axis Theorem | ||

+ | #* Evaluate the mass and center or mass of a plate | ||

+ | #* Evaluate the centroid of a plate. | ||

+ | #* Evaluate the moments of inertia of a plate. | ||

+ | #* Calculate the radius of gyration of a plate. | ||

+ | #* Recall and apply the parallel axis theorem. | ||

+ | # Triple Integrals; Reduction to Repeated Integrals | ||

+ | #* Define the following: | ||

+ | #** triple integral | ||

+ | #** integral formula for the volume of a solid | ||

+ | #** integral formula for the mass of a solid | ||

+ | #** integral formulas for the center of mass of a solid | ||

+ | #* Evaluate physical quantities using triple integrals such as volume, mass, center of mass, and moments of intertia. | ||

+ | #* Recall and apply the properties of triple integrals, including: linearity, order, additivity, and the mean-value condition. | ||

+ | #* Sketch the domain of integration of an iterated integral. | ||

+ | #* Change the order of integration of a triple integral. | ||

+ | # Cylindrical Coordinates | ||

+ | #* Define the following: | ||

+ | #** cylindrical coordinates of a point | ||

+ | #** coordinate transformations between Cartesian and cylindrical coordinates | ||

+ | #** cylindrical element of volume | ||

+ | #* Convert between Cartesian and cylindrical coordinates. | ||

+ | #* Describe regions in cylindrical coordinates. | ||

+ | #* Evaluate triple integrals using cylindrical coordinates. | ||

+ | # Spherical Coordinates | ||

+ | #* Define the following: | ||

+ | #** spherical coordinates of a point | ||

+ | #** coordinate transformations between Cartesian and spherical coordinates | ||

+ | #** spherical element of volume | ||

+ | #* Convert between Cartesian and spherical coordinates. | ||

+ | #* Describe regions in spherical coordinates. | ||

+ | #* Evaluate triple integrals using spherical coordinates. | ||

+ | # Jacobians; Changing Variables in Multiple Integration | ||

+ | #* Define the following: | ||

+ | #** Jacobian | ||

+ | #** change of variable formula for double integration | ||

+ | #** change of variable formula for triple integration | ||

+ | #* Find the Jacobian of a coordinate transformation. | ||

+ | #* Use a coordinate transformation to evaluate double and triple integrals. | ||

+ | # Line Integrals | ||

+ | #* Define the following: | ||

+ | #** work along a curved path | ||

+ | #** smooth parametric curve | ||

+ | #** directed or oriented curve | ||

+ | #** path dependence | ||

+ | #** closed curve | ||

+ | #* Evaluate the work done by a varying force over a curved path. | ||

+ | #* Evaluate line integrals in general including line integrals with respect to arc length. | ||

+ | #* Evaluate the physical characteristics of a wire such as centroid, mass, and center of mass using line integrals. | ||

+ | #* Determine whether or not a vector field is a gradient. | ||

+ | #* Determine whether or not a differential form is exact. | ||

+ | # The Fundamental Theorem for Line Integrals; Work-Energy Formula; Conservation of Mechanical Energy | ||

+ | #* Define the following: | ||

+ | #** path-independent line integrals | ||

+ | #** closed vector field | ||

+ | #** simply connected | ||

+ | #* Recall, apply, and verify the Fundamental Theorem for Line Integrals (Theorem 2 in Section 15.3). | ||

+ | #* Determine whether or not a force field is closed on a given region, and if so, find its potential function. | ||

+ | #* Solve application problems involving work done by a conservative vector field | ||

+ | # Vector Fields | ||

+ | #* Define the following: | ||

+ | #** vector field | ||

+ | #** open | ||

+ | #** path connected | ||

+ | #** region | ||

+ | #** integral curve (field lines, flow lines, or streamlines) | ||

+ | #** gradient vector field (or conservative vector field) | ||

+ | #** potential function | ||

+ | #** continuously differentiable vector field | ||

+ | #* Sketch a vector field. | ||

+ | #* Write the formula for a vector field from a description. | ||

+ | #* Write the gradient vector field associated with a given scalar-valued function. | ||

+ | #* Recover a function from its gradient or show it is not possible. | ||

+ | #* Find the integral curves of a vector field. | ||

+ | # Green's Theorem | ||

+ | #* Define the following: | ||

+ | #** Jordan curve | ||

+ | #** Jordan region | ||

+ | #** Green's Theorem | ||

+ | #* Recall and verify Green's Theorem. | ||

+ | #* Apply Green's Theorem to evaluate line integrals. | ||

+ | #* Apply Green's Theorem to find the area of a region. | ||

+ | #* Derive identities involving Green's Theorem | ||

+ | # Parameterized Surfaces; Surface Area | ||

+ | #* Define the following: | ||

+ | #** parameterized surface | ||

+ | #** fundamental vector product | ||

+ | #** element of surface area for a parameterized surface | ||

+ | #** surface integral | ||

+ | #** integral formula for the surface area of a parameterized surface | ||

+ | #** integral formula for the surface area of a surface z = f(x; y) | ||

+ | #** upward unit normal | ||

+ | #* parameterize a surface. | ||

+ | #* evaluate the fundamental vector product for a parameterized surface. | ||

+ | #* Calculate the surface area of a parameterized surface. | ||

+ | #* Calculate the surface area of a surface z = f(x; y). | ||

+ | # Surface Integrals | ||

+ | #* Define the following: | ||

+ | #** surface integral | ||

+ | #** integral formulas for the surface area and centroid of a parameterized surface | ||

+ | #** integral formulas for the mass and center of mass of a parameterized surface | ||

+ | #** integral formulas for the moments of inertia of a parameterized surface | ||

+ | #** integral formula for flux through a surface | ||

+ | #* Calculate the surface area and centroid of a parameterized surface. | ||

+ | #* Calculate the mass and center of mass of a parameterized surface. | ||

+ | #* Calculate the moments of inertia of a parameterized surface. | ||

+ | #* Evaluate the flux of a vector field through a surface. | ||

+ | #* Solve application problems involving surface integrals. | ||

+ | # The Vector Differential Operator Del | ||

+ | #* Define the following: | ||

+ | #** the vector differential operator Del | ||

+ | #** divergence | ||

+ | #** curl | ||

+ | #** Laplacian | ||

+ | #* Evaluate the divergence of a vector field. | ||

+ | #* Evaluate the curl of a vector field | ||

+ | #* Evaluate the Laplacian of a function. | ||

+ | #* Recall, derive and apply formulas involving divergence, gradient and Laplacian. | ||

+ | #* Interpret that divergence and curl of a vector fields physically. | ||

+ | # The Divergence Theorem | ||

+ | #* Define the following: | ||

+ | #** outward unit normal | ||

+ | #** the divergence theorem | ||

+ | #** sink and source | ||

+ | #** solenoidal | ||

+ | #* Recall and verify the Divergence Theorem. | ||

+ | #* Apply the Divergence Theorem to evaluate the flux through a surface. | ||

+ | #* Solve application problems using the Divergence Theorem. | ||

+ | # Stokes' Theorem | ||

+ | #* Define the following: | ||

+ | #** oriented surface | ||

+ | #** outward, upward, and downward unit normal | ||

+ | #** the positive sense around the boundary of a surface | ||

+ | #** circulation | ||

+ | #** component of curl in the normal direction | ||

+ | #** irrotational | ||

+ | #** Stokes' theorem | ||

+ | #* Recall and verify Stoke's theorem. | ||

+ | #* Use Stokes' Theorem to calculate the flux of a curl vector field through a surface by a line integral. | ||

+ | #* Apply Stokes' theorem to calculate the work (or circulation) of a vector field around a simple closed curve. | ||

+ | |||

+ | |||

+ | |||

</div> | </div> | ||

Line 29: | Line 737: | ||

=== Courses for which this course is prerequisite === | === Courses for which this course is prerequisite === | ||

+ | [[Math 303]] | ||

[[Category:Courses|302]] | [[Category:Courses|302]] |

## Revision as of 14:47, 17 February 2010

## Contents

## Catalog Information

### Title

Mathematics for Engineering 1.

### (Credit Hours:Lecture Hours:Lab Hours)

(4:4:0)

### Offered

F, W

### Prerequisite

Math 113 and passing grade on required preparatory exam taken during first week of class. (Practice exams available on class website.)

### Description

Multivariable calculus, linear algebra, and numerical methods.

## Desired Learning Outcomes

This course is designed to give students from the College of Engineering and Technology the mathematics background necessary to succeed in their chosen field.

### Prerequisites

Students are expected to have completed Math 113.

### Minimal learning outcomes

- Rectangular Space Coordinates; Vectors in Three-Dimensional Space
- Define the following:
- Cartesian coordinates of a point
- sphere
- symmetry about a point, a line, and a plane
- vector
- components of a vector
- vector addition
- scalar multiplication
- zero vector
- vector subtraction
- vector norm (magnitude, length)
- unit vector
- coordinate unit vectors i, j, k
- linear combination of unit vectors

- Plot points in three-dimensional space.
- Calculate the distance between two points in two-dimensional space and 3-dimensional space
- Write the equation of a sphere centered about a given point with a given radius. Determine the center and radius of a sphere, given its equation.
- Write the component equations of a line that passes through two given points.
- Write the component equations of a line segment with given endpoints.
- Find the midpoint of a given line segment.
- Find the points of symmetry about a point, line, or plane.
- Represent a vector by each of the following:
- components
- a linear combination of coordinate unit vectors

- Carry out the vector operations:
- addition
- scalar multiplication
- subtraction

- Represent the operations of vector addition, scalar multiplication and norm geometrically.
- Find the norm (magnitude, length) of a vector. Determine whether two vectors are parallel.
- Recall, apply and verify the basic properties of vector addition, scalar multiplication and norm.
- Model and solve application problems using vectors.

- Define the following:
- The Dot Product
- Define the following:
- dot product.
- perpendicular vectors.
- unit vector in the direction of a vector a, denoted u_a.
- the projection of a on b, denoted proj_b a.
- the b-component of a, denoted comp_b a.
- the direction cosines of a vector.
- the direction angles of a vector.
- the Schwarz Inequality.
- the work done by a constant force on an object.
- the dot product test for perpendicular vectors.
- the dot product test for parallel vectors.
- geometric interpretation of the dot product

- Evaluate a dot product from the coordinate formula or the angle formula.
- Interpret the dot product geometrically.
- Evaluate the following using the dot product:
- the length of a vector.
- the angle between two vectors.
- u_a, the unit vector in the direction of a vector a.
- proj_b a, the projection of a on b.
- comp_b a, the b-component of a.
- the direction cosines of a vector.
- the direction angles of a vector.
- the work done by a constant force on an object.

- Prove and verify the Schwarz Inequality.
- Prove and apply the dot product tests for perpendicular and parallel vectors.
- Recall and apply the properties of the dot product.
- Prove identities involving the dot product.
- Solve application problems involving the dot product.
- Extend the vector operations and related identities for addition, scalar multiplication, and dot product to higher dimensions.

- Define the following:
- The Cross Product
- Define the following:
- the cross product of two vectors
- scalar triple product

- Evaluate a cross product from the the coordinate formula or angle formula.
- Interpret the cross product geometrically.
- Evaluate the following using the cross product:
- a vector perpendicular to two given vectors.
- the area of a parallelogram.
- the area or a triangle.
- moment of force or moment of torque.

- Evaluate scalar triple products.
- Use the scalar triple product to determine the following:
- volume of a parallelepiped.
- whether or not three vectors are coplanar.

- Recall and apply the properties of the cross product and scalar triple product.
- Prove identities involving the cross product and the scalar triple product.
- Solve application problems involving the cross product and scalar triple product.

- Define the following:
- Lines
- Define the following:
- direction vector for a line
- vector equation of a line
- scalar parametric equations of a line
- Cartesian equations or symmetric form of a line

- Represent a line in 3-space by:
- a vector equation
- scalar parametric equations
- Cartesian equations

- Find the equation(s) representing a line given information about
- a point of the line and the direction of the line or
- two points contained in the line.
- a point and a parallel line.
- a point and perpendicular to a plane.
- two planes intersecting in the line.

- Find the distance from a point to a line.
- Solve application problems involving lines.

- Define the following:
- Planes
- Define the following:
- normal vector to a plane
- cartesian equation of a plane
- parametric equation of a plane

- Find the equation of a plane in 3-space given a point and a normal vector, three points, or a geometric description of the plane.
- Determine a normal vector and the intercepts of a given plane.
- Represent a plane by parametric equations.
- Find the distance from a point to a plane.
- Find the angle between a line and a plane.
- Determine a point of intersection between a line and a surface.
- Sketch planes given their equations.
- Solve application problems involving planes.

- Define the following:
- Systems of Linear Equations
- Define the following:
- linear system of m equations in n unknowns
- consistent and inconsistent
- solution set
- coefficient matrix
- elementary row operations

- Identify linear systems.
- Represent a system of linear equations as an augmented matrix and vice versa.
- Relate the following types of solution sets of a system of two or three variables to the intersections of lines in a plane or the intersection of planes in three space:
- a unique solution.
- infinitely many solutions.
- no solution.

- Define the following:
- Gaussian elimination
- Define the following:
- reduced row echelon form
- leading variables or pivots
- free variables
- row echelon form
- back substitution
- Gaussian elimination
- Gauss-Jordan elimination
- homogeneous
- trivial solution
- nontrivial solutions

- Identify matrices that are in row echelon form and reduced row echelon form.
- Determine whether a linear system is consistent or inconsistent from its reduced row echelon form. If the system is consistent, write the solution.
- Identify the lead variables and free variables of a system represented by an augmented matrix in reduced row echelon form.
- Solve systems of linear equations using Gaussian elimination and back substitution.
- Solve systems of linear equations using Gauss-Jordan elimination.
- Model and solve application problems using linear systems.

- Define the following:
- Matrices and Matrix Operations
- Define the following:
- vector, row vector, and column vector
- equal matrices
- scalar multiplication
- sum of matrices
- zero matrix
- scalar product
- linear combination
- matrix multiplication
- transpose
- trace
- identity matrix

- Perform the operations of matrix addition, scalar multiplication, transposition, trace, and matrix multiplication.
- Represent matrices in terms of double subscript notation.

- Define the following:
- Inverses; Rules of Matrix Arithmetic
- Define the following:
- commutative property
- singular
- nonsingular or invertible
- multiplicative inverse

- Recall, demonstrate, and apply algebraic properties for matrices.
- Recall that matrix multiplication is not commutative in general. Determine conditions under which matrices do commute.
- Recall and prove properties and identities involving the transpose operator.
- Recall and prove properties and identities involving matrix inverses.
- Recall and prove properties and identities involving matrix powers.
- Recall, demonstrate, and apply that the cancelation laws for scalar multiplication do not hold for matrix multiplication.
- Recall and apply the formula for the inverse of 2x2 matrices.

- Define the following:
- Elementary Matrices
- Define the following:
- elementary matrix
- row equivalent matrices

- Identify elementary matrices and find their inverses or show that their inverse does not exist.
- Relate elementary matrices to row operations.
- Factor matrices using elementary matrices.
- Find the inverse of a matrix, if possible, using elementary matrices.
- Prove theorems about matrix products and matrix inverses.
- Solve a linear equation using matrix inverses.

- Define the following:
- Further Results on Systems of Equations and Invertibility
- Solve matrix equations using matrix algebra.
- Recall and prove properties and identities involving matrix inverses.
- Recall equivalent conditions for invertibility.

- Further Results on Systems of Equations and Invertibility
- Define the following:
- diagonal matrix
- upper and lower triangular matrices
- symmetric matrix
- skew-symmetric matrix

- Determine powers of diagonal matrices.
- Recall and prove properties and identities involving the transpose operator.
- Prove basic facts involving symmetric and skew-symmetric matrices.

- Define the following:
- Determinants
- Define the following:
- minor
- cofactor
- cofactor expansion
- determinant
- adjoint
- Cramer's Rule

- Apply cofactor expansion to evaluate determinants of nxn matrices.
- Recall and apply the properties of determinants to evaluate determinants.
- Evaluate the adjoint of a matrix.
- Determine whether or not a matrix has an inverse based on its determinant.
- Evaluate the inverse of a matrix using the adjoint method.
- Use Cramer's rule to solve a linear system.

- Define the following:
- Properties of Determinants
- Recall the effects that row operations have on the determinants of matrices. Relate to the determinants of elementary matrices.
- Recall, apply and verify the properties of determinants to evaluate determinants, including:
- det(AB) = det(A) det(B)
- det(kA) = k^n det(A)
- det(A^-1)= 1/det(A)
- det(A^T) = det(A)
- det(A) = 0 if and only if A is singular

- Evaluate the determinant of a matrix using row operations.
- Apply determinants to determine invertibility of matrix products.

- Linear Transformations: Definitions and Examples
- Define the following:
- linear transformation
- image
- range

- Describe geometrically the effects of a linear operator.
- Determine whether or not a given transformation is linear.
- Prove theorems and solve application problems involving linear transformations.

- Define the following:
- Matrix Representations of Linear Transformations
- Define the following:
- standard matrix representation
- eigenvalues and eigenvectors

- Determine the matrix that represents a given linear transformation of vectors given an algebraic description.
- Determine the matrix that represents a given linear transformation of vectors given a geometric description.
- Prove theorems and solve application problems involving linear transformations.

- Define the following:
- Vector Spaces: Definitions and Examples
- Define the following:
- vector space
- vector space axioms
- vector space R^n
- vector space R^(mxn)
- vector space of real-valued functions
- additional properties of vector spaces

- Prove or disprove that a given set of vectors together with an addition and a scalar multiplication is a vector space.
- Prove and verify properties of a vector space.

- Define the following:
- Subspaces
- Define the following:
- subspace
- closure under addition
- closure under scalar multiplication
- zero subspace
- linear combination
- span (or subspace spanned by a set of vectors)
- spanning set

- Prove or disprove that a set of vectors forms a subspace.
- Prove or disprove a set of vectors is a spanning set for R^n.
- Prove or disprove a given vector is in the span of a set of vectors. Determine the span of a set of vectors.
- Prove theorems about vector spaces and spans.

- Define the following:
- Linear Independence
- Define the following:
- linearly independent
- linearly dependent
- Wronskian

- Determine whether a set of vectors is linearly dependent or linearly independent.
- Geometrically describe the span of a set of vectors. For sets that are linearly dependent, determine a dependence relation.
- Prove theorems about linear independence.

- Define the following:
- Basis and Dimension
- Define the following:
- basis
- dimension
- finite and infinite dimensional
- standard basis

- Prove or disprove a set of vectors forms a basis.
- Find a basis for a vector space.
- Determine the dimension of a vector space.
- Geometrically interpret the ideas of span, linear dependance, basis, and dimension.

- Define the following:
- Row Space, Column Space, and Null Space
- Define the following:
- row space
- column space
- null space
- particular solution
- general solution

- Express a product Ax as a linear combination of column vectors.
- Find a basis for a the column space, the row space, and the null space of a matrix.
- Find the basis for a span of vectors.

- Define the following:
- Rank and Nullity
- Define the following:
- rank
- nullity
- The Consistency Theorem
- equivalent statements of invertibility

- Find the rank and nullity of a matrix.
- Recall and prove identities involving rank and nullity
- Recall and apply the Consistency Theorm
- Recall and apply the equivalent statements of invertibility.

- Define the following:
- Eigenvalues and Eigenvectors
- Define the following:
- eigenvalue or characteristic value
- eigenvector or characteristic vector
- characteristic polynomial or characteristic polynomial
- equivalent statements of invertibility

- Find the eigenvalues and eigenvectors of an nxn matrix.
- Prove theorems and solve application problems involving eigenvalues and eigenvectors.

- Define the following:
- Diagonalization
- Define the following:
- diagonalizable
- algebraic multiplicity
- geometric multiplicity

- Determine whether or not a matrix is diagonalizable.
- Find the diagonalization of a matrix, if possible.
- Find powers of a matrix using the diagonalization of a matrix.
- Prove theorems and solve application problems involving the diagonalization of matrices.

- Define the following:
- Limit, Continuity, Vector Derivative; The Rules of Differentiation
- Define the following:
- scalar functions
- vector functions
- components of a vector function
- plane curve or space curve
- parametrization of a curve
- limit of a vector function
- a vector function continuous at a point
- derivative of a vector function
- a differentiable vector function
- integral of a vector function

- Graph a parametric curve.
- Identify a curve given its parametrization.
- Determine combinations of vector functions such as sums, vector products and scalar products.
- Evaluate limits, derivatives, and integrals of vector functions.
- Recall, derive and apply rules to combinations of vector functions for the following:
- limits
- differentiation
- integration

- Determine continuity of a vector-valued function.
- Prove theorems involving limits and derivatives of vector-valued functions.
- Solve application problems involving vector-valued functions.

- Define the following:
- Curves; Vector Calculus in Mechanics
- Define the following:
- directed path
- differentiable parameterized curve
- tangent vector
- tangent line
- unit tangent vector
- principal normal vector
- normal line
- osculation plane
- force vector
- momentum vector
- angular momentum vector
- torque

- Find the tangent vector and tangent line to a curve at a given point.
- Find the principle normal and normal line to a curve at a given point.
- Determine the osculating plane for a space curve at a given point.
- Reverse the direction of a curve.
- Solve application problems involving curves.
- Solve application problems involving force, momentum, angular momentum, and torque.

- Define the following:
- Arc Length
- Define the following:
- arc length
- arc length parametrization

- Evaluate the arc length of a curve.
- Determine whether a curve is arc length parameterized.
- Find the arc length parametrization of a curve.

- Define the following:
- Curvilinear Motion; Curvature
- Define the following:
- velocity vector function
- speed
- acceleration vector function
- uniform circular motion
- curvature
- tangential component of acceleration
- normal component of acceleration

- Given the position vector function of a moving object, calculate the velocity vector function, speed, and acceleration vector function, and vice versa.
- Calculate the curvature of a space curve.
- Recall the formulas for the curvature of a parameterized planar curve or a planar curve that is the graph of a function. Apply these formulas to calculate the curvature of a planar curve.
- Determine the tangential and normal components of acceleration for a given parameterized curve.
- Solve application problems involving curvilinear motion and curvature.

- Define the following:
- Functions of Several Variables; A Brief Catalogue of the Quadric Surfaces; Projections
- Define the following:
- real-valued function of several variables
- domain
- range
- bounded functions
- quadric surface
- intercepts
- traces
- sections
- center
- symmetry
- boundedness
- cylinder
- ellipsiod
- elliptic cone
- elliptic paraboloid
- hyperboloid of one sheet
- hyperboloid of two sheets
- hyperbolic paraboloid
- parabolic cylinder
- elliptic cylinder
- projection of a curve onto a coordinate plane

- Describe the domain and range of a function of several variables.
- Write a function of several variables given a description.
- Identify standard quadratic surfaces given their functions or graphs.
- Sketch the graph of a quadratic surface by sketching intercepts, traces, sections, centers, symmetry, boundedness.
- Find the projection of a curve, that is the intersection of two surfaces, to a coordinate plane.

- Define the following:
- Graphs; Level Curves and Level Surfaces
- Define the following:
- level curve
- level surface

- Describe the level sets of a function of several variables.
- Graphically represent a function of two variables by level curves or a function of three variables by level surfaces.
- Identify the characteristics of a function from its graph or from a graph of its level curves (or level surfaces).
- Solve application problems involving level sets. functions.

- Define the following:
- Partial Derivatives
- Define the following:
- partial derivative of a function of several variables
- second partial derivative
- mixed partial derivative

- Interpret the definition of a partial derivative of a function of two variables graphically.
- Evaluate the partial derivatives of a function of several variables.
- Evaluate the higher order partial derivatives of a function of several variables.
- Verify equations involving partial derivatives.
- Apply partial derivatives to solve application problems.

- Define the following:
- Open and Closed Sets; Limits and Continuity; Equity of Mixed Partials
- Define the following:
- neighborhood of a point
- deleted neighborhood of a point
- interior of a set
- boundary of a set
- open set
- closed set
- limit of a function of several variables at a point
- continuity of a function of several variables at a point

- Determine the boundary and interior of a set.
- Determine whether a set is open, closed, neither, or both.
- Evaluate the limit of a function of several variables or show that it does not exists.
- Determine whether or not a function is continuous at a given point.
- Recall and apply the conditions under which mixed partial derivatives are equal.

- Define the following:
- Differentiability and Gradient
- Define the following:
- differentiable multivariable function
- gradient of a multivariable function

- Evaluate the gradient of a function.
- Find a function with a given gradient.

- Define the following:
- Gradient and Directional Derivative
- Define the following:
- directional derivative
- isothermals

- Recall and prove identities involving gradients.
- Give a graphical interpretation of the gradient.
- Evaluate the directional derivative of a function.
- Give a graphical interpretation of directional derivative.
- Recall, prove, and apply the theorem that states that a differential function f increases most rapidly in the direction of the gradient (the rate of change is then ||f(x)||) and it decreases most rapidly in the opposite direction (the rate of change is then -||f(x)||).
- Find the path of a heat seeking or a heat repelling particle.
- Solve application problems involving gradient and directional derivatives.

- Define the following:
- The Mean-Value Theorem; The Chain Rule
- Define the following:
- the Mean Value Theorem for functions of several variables
- normal line
- chain rules for functions of several variables
- implicit differentiation

- Recall and apply the Mean Value Theorem for functions of several variables and its corollaries.
- Apply an appropriate chain rule to evaluate a rate of change.
- Apply implicit differentiation to evaluate rates of change.
- Solve application problems involving chain rules and implicit differentiation.

- Define the following:
- The Gradient as a Normal; Tangent Lines and Tangent Planes
- Define the following:
- normal vector
- tangent vector
- tangent line
- tangent plane
- normal line

- Use gradients to find the normal vector and normal line to a smooth planar curve at a given point.
- Use gradients to find the tangent vector and tangent line to a smooth planar curve at a given point.
- Use gradients to find the normal vector to a smooth surface at a given point.
- Use gradients to find the tangent plane to a smooth surface at a given point.
- Use gradients to find the normal line to a smooth surface at a given point.
- Solve application problems involving normals and tangents to curves and surfaces.

- Define the following:
- Local Extreme Values
- Define the following:
- local minimum and local maximum
- critical points
- stationary points
- saddle points
- discriminant
- Second Derivative Test

- Find the critical points of a function of two variables.
- Apply the Second-Partials Test to determine whether each critical point is a local minimum, a local maximum, or a saddle point.
- Solve word problems involving local extreme values.

- Define the following:
- Absolute Extreme Values
- Define the following:
- absolute minimum and absolute maximum
- bounded subset of a plane or three-space
- the Extreme Value Theorem

- Determine absolute extreme values of a function defined on a closed and bounded set.
- Apply the Extreme Value Theorem to justify the method for finding extreme values of functions defined on certain sets.
- Solve word problems involving absolute extreme values.

- Define the following:
- Maxima and Minima with Side Conditions
- Define the following:
- side conditions or constraints
- method of Lagrange
- Lagrange multipliers
- cross-product equation of the Lagrange condition

- Graphically interpret the method of Lagrange.
- Determine the extreme values of a function subject to a side conditions by applying the method of Lagrange.
- Apply the cross-product equation of the Lagrange condition to solve extreme value problems subject to side conditions.
- Apply the method of Lagrange to solve word problems.

- Define the following:
- Differentials; Reconstructing a Function from its Gradient
- Define the following:
- differential
- general solution
- particular solution
- connected open set
- open region
- simple closed curve
- simply connected open region
- partial derivative gradient test

- Determine the differential for a given function of several variables.
- Determine whether or not a vector function is a gradient.
- Given a vector function that is a gradient, find the functions with that gradient.

- Define the following:
- Multiple-Sigma Notation; The Double Integral over a Rectangle R; The Evaluation of Double Integrals by Repeated Integrals
- Define the following:
- double sigma notation
- triple sigma notation
- upper sum
- lower sum
- double integral
- integral formula for the volume of a solid bounded between a region Omega in the xy-plane and the graph of a non-negative function z = f(x,y) defined on Omega.
- integral formula for the area of region in a plane
- integral formula for the average of a function defined on a region Omega.
- projection of a region onto a coordinate axis
- Type I and Type II regions
- reduction formulas for double integrals
- the geometric interpretation of the reduction formulas for double integrals

- Evaluate double and triple sums given their sigma notation.
- Recall and apply summation identities.
- Approximate the integral of a function by a lower sum and an upper sum.
- Evaluate the integral of a function using the definition.
- Evaluate double integrals over a rectangle using the reduction formulas.
- Sketch planar regions and determine if they are Type I, Type II, or both.
- Evaluate double integrals over Type I and Type II regions.
- Change the order of integration of an integral.
- Apply double integrals to calculate volumes, areas, and averages.

- Define the following:
- The Double Integral as the Limit of Riemann Sums; Polar Coordinates
- Define the following:
- diameter of a set
- Riemann sum
- double integral as a limit of Riemann sums
- polar coordinates (r; theta)
- transformation formulas between Cartesian and polar coordinates
- double integral conversion formula between Cartesian and polar coordinates

- Represent a region in both Cartesian and polar coordinates.
- Evaluate double integrals in terms of polar coordinates.
- Evaluate areas and volumes using polar coordinates.
- Convert a double integral in Cartesian coordinates to a double integral in polar coordinates and then evaluate.

- Define the following:
- Further Applications of the Double Integral
- Define the following:
- integral formula for the mass of a plate
- integral formulas for the center of mass of a plate
- integral formulas for the centroid of a plate
- integral formulas for the moment of an inertia of a plate
- radius of gyration
- the Parallel Axis Theorem

- Evaluate the mass and center or mass of a plate
- Evaluate the centroid of a plate.
- Evaluate the moments of inertia of a plate.
- Calculate the radius of gyration of a plate.
- Recall and apply the parallel axis theorem.

- Define the following:
- Triple Integrals; Reduction to Repeated Integrals
- Define the following:
- triple integral
- integral formula for the volume of a solid
- integral formula for the mass of a solid
- integral formulas for the center of mass of a solid

- Evaluate physical quantities using triple integrals such as volume, mass, center of mass, and moments of intertia.
- Recall and apply the properties of triple integrals, including: linearity, order, additivity, and the mean-value condition.
- Sketch the domain of integration of an iterated integral.
- Change the order of integration of a triple integral.

- Define the following:
- Cylindrical Coordinates
- Define the following:
- cylindrical coordinates of a point
- coordinate transformations between Cartesian and cylindrical coordinates
- cylindrical element of volume

- Convert between Cartesian and cylindrical coordinates.
- Describe regions in cylindrical coordinates.
- Evaluate triple integrals using cylindrical coordinates.

- Define the following:
- Spherical Coordinates
- Define the following:
- spherical coordinates of a point
- coordinate transformations between Cartesian and spherical coordinates
- spherical element of volume

- Convert between Cartesian and spherical coordinates.
- Describe regions in spherical coordinates.
- Evaluate triple integrals using spherical coordinates.

- Define the following:
- Jacobians; Changing Variables in Multiple Integration
- Define the following:
- Jacobian
- change of variable formula for double integration
- change of variable formula for triple integration

- Find the Jacobian of a coordinate transformation.
- Use a coordinate transformation to evaluate double and triple integrals.

- Define the following:
- Line Integrals
- Define the following:
- work along a curved path
- smooth parametric curve
- directed or oriented curve
- path dependence
- closed curve

- Evaluate the work done by a varying force over a curved path.
- Evaluate line integrals in general including line integrals with respect to arc length.
- Evaluate the physical characteristics of a wire such as centroid, mass, and center of mass using line integrals.
- Determine whether or not a vector field is a gradient.
- Determine whether or not a differential form is exact.

- Define the following:
- The Fundamental Theorem for Line Integrals; Work-Energy Formula; Conservation of Mechanical Energy
- Define the following:
- path-independent line integrals
- closed vector field
- simply connected

- Recall, apply, and verify the Fundamental Theorem for Line Integrals (Theorem 2 in Section 15.3).
- Determine whether or not a force field is closed on a given region, and if so, find its potential function.
- Solve application problems involving work done by a conservative vector field

- Define the following:
- Vector Fields
- Define the following:
- vector field
- open
- path connected
- region
- integral curve (field lines, flow lines, or streamlines)
- gradient vector field (or conservative vector field)
- potential function
- continuously differentiable vector field

- Sketch a vector field.
- Write the formula for a vector field from a description.
- Write the gradient vector field associated with a given scalar-valued function.
- Recover a function from its gradient or show it is not possible.
- Find the integral curves of a vector field.

- Define the following:
- Green's Theorem
- Define the following:
- Jordan curve
- Jordan region
- Green's Theorem

- Recall and verify Green's Theorem.
- Apply Green's Theorem to evaluate line integrals.
- Apply Green's Theorem to find the area of a region.
- Derive identities involving Green's Theorem

- Define the following:
- Parameterized Surfaces; Surface Area
- Define the following:
- parameterized surface
- fundamental vector product
- element of surface area for a parameterized surface
- surface integral
- integral formula for the surface area of a parameterized surface
- integral formula for the surface area of a surface z = f(x; y)
- upward unit normal

- parameterize a surface.
- evaluate the fundamental vector product for a parameterized surface.
- Calculate the surface area of a parameterized surface.
- Calculate the surface area of a surface z = f(x; y).

- Define the following:
- Surface Integrals
- Define the following:
- surface integral
- integral formulas for the surface area and centroid of a parameterized surface
- integral formulas for the mass and center of mass of a parameterized surface
- integral formulas for the moments of inertia of a parameterized surface
- integral formula for flux through a surface

- Calculate the surface area and centroid of a parameterized surface.
- Calculate the mass and center of mass of a parameterized surface.
- Calculate the moments of inertia of a parameterized surface.
- Evaluate the flux of a vector field through a surface.
- Solve application problems involving surface integrals.

- Define the following:
- The Vector Differential Operator Del
- Define the following:
- the vector differential operator Del
- divergence
- curl
- Laplacian

- Evaluate the divergence of a vector field.
- Evaluate the curl of a vector field
- Evaluate the Laplacian of a function.
- Recall, derive and apply formulas involving divergence, gradient and Laplacian.
- Interpret that divergence and curl of a vector fields physically.

- Define the following:
- The Divergence Theorem
- Define the following:
- outward unit normal
- the divergence theorem
- sink and source
- solenoidal

- Recall and verify the Divergence Theorem.
- Apply the Divergence Theorem to evaluate the flux through a surface.
- Solve application problems using the Divergence Theorem.

- Define the following:
- Stokes' Theorem
- Define the following:
- oriented surface
- outward, upward, and downward unit normal
- the positive sense around the boundary of a surface
- circulation
- component of curl in the normal direction
- irrotational
- Stokes' theorem

- Recall and verify Stoke's theorem.
- Use Stokes' Theorem to calculate the flux of a curl vector field through a surface by a line integral.
- Apply Stokes' theorem to calculate the work (or circulation) of a vector field around a simple closed curve.

- Define the following: