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

### Title

Mathematics for Engineering 1.

(4:4:0)

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

1. 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
• 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:
• 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.
2. 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.
3. 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.
4. 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.
5. 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.
6. 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.
7. Gaussian elimination
• Define the following:
• reduced row echelon form
• 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.
8. 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.
9. 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.
10. 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.
11. 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.
12. 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.
13. Determinants
• Define the following:
• minor
• cofactor
• cofactor expansion
• determinant
• 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.
14. 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.
15. 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.
16. 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.
17. 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.
18. Subspaces
• Define the following:
• subspace
• 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.
19. 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.
20. 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.
21. 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.
22. 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.
23. 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.
24. 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.
25. 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.
26. 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.
27. 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.
28. 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.
29. 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
• 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.
30. 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.
31. 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.
32. 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:
• differentiable multivariable function
• gradient of a multivariable function
• Evaluate the gradient of a function.
• Find a function with a given gradient.
• 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.
35. 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.
36. 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.
37. Local Extreme Values
• Define the following:
• local minimum and local maximum
• critical points
• stationary 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.
38. 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.
39. 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.
40. 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
• 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.
41. 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.
42. 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.
43. 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
• 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.
44. 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.
45. 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.
46. 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.
47. 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.
48. 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.
49. 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
50. 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.
51. 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
52. 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).
53. 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.
54. 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.
55. 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.
56. 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.

### Textbooks

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