# Math 302: Math for Engineering 1

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## Contents

## Catalog Information

### Title

Mathematics for Engineering 1.

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

(4:4:0)

### Offered

F, W

### Prerequisite

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

### Description

Multivariable calculus, linear algebra, and numerical methods.

## Desired Learning Outcomes

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

### Prerequisites

Students are expected to have completed Math 113.

### Minimal learning outcomes

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

- Define the following:

### Textbooks

Possible textbooks for this course include (but are not limited to):