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

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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="-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>
 
 
=== Textbooks ===
 
Possible textbooks for this course include (but are not limited to):
 
 
*
 
 
 
=== Additional topics ===
 
 
=== Courses for which this course is prerequisite ===
 
[[Math 303 Mathematics for Engineering 2|Math 303]]
 
 
[[Category:Courses|302]]
 

Latest revision as of 15:14, 3 April 2013

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