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−  == 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 ===
 
− 
 
−  <div style="mozcolumncount:2; columncount:2;">
 
− 
 
−  # Rectangular Space Coordinates; Vectors in ThreeDimensional 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 threedimensional space.
 
−  #* Calculate the distance between two points in twodimensional space and 3dimensional 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 bcomponent 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 bcomponent 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 3space 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 3space 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
 
−  #** GaussJordan 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 GaussJordan 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
 
−  #** skewsymmetric matrix
 
−  #* Determine powers of diagonal matrices.
 
−  #* Recall and prove properties and identities involving the transpose operator.
 
−  #* Prove basic facts involving symmetric and skewsymmetric 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 realvalued 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 vectorvalued function.
 
−  #* Prove theorems involving limits and derivatives of vectorvalued functions.
 
−  #* Solve application problems involving vectorvalued 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:
 
−  #** realvalued 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 MeanValue 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 SecondPartials 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 threespace
 
−  #** 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
 
−  #** crossproduct 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 crossproduct 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.
 
−  # MultipleSigma 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 xyplane and the graph of a nonnegative 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 meanvalue 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; WorkEnergy Formula; Conservation of Mechanical Energy
 
−  #* Define the following:
 
−  #** pathindependent 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 scalarvalued 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.
 
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−  </div>
 
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−  === Textbooks ===
 
−  Possible textbooks for this course include (but are not limited to):
 
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−  *
 
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−  === Additional topics ===
 
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−  === Courses for which this course is prerequisite ===
 
−  [[Math 303 Mathematics for Engineering 2Math 303]]
 
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−  [[Category:Courses302]]
 