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

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== Catalog Information ==
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{{db-g7}}
 
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=== Title ===
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Mathematics for Engineering 1.
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=== (Credit Hours:Lecture Hours:Lab Hours) ===
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(4:4:0)
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=== Offered ===
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F, W
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=== Prerequisite ===
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[[Math 113]] and passing grade on required preparatory exam taken during first week of class. (Practice exams available on class website.)
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=== Description ===
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Multivariable calculus, linear algebra, and numerical methods.
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== Desired Learning Outcomes ==
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This course is designed to give students from the College of Engineering and Technology the mathematics background necessary to succeed in their chosen field.
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=== Prerequisites ===
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Students are expected to have completed [[Math 113]].
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=== Minimal learning outcomes ===
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<div style="-moz-column-count:2; column-count:2;">
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# 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
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#** 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.
+
 
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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 2|Math 303]]
+
 
+
[[Category:Courses|302]]
+

Latest revision as of 16:14, 3 April 2013

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