## Vector Fields

- Definition of a Vector Field
- Physical Interpretation of Vector Fields
- Example #1 sketch a sample Vector Field
- Example #2 sketch a Gradient Vector Field
- Example #3 Sketch a Gradient Vector Field
- Two Examples of how to find the Gradient Vector Field
- Overview of Conservative Vector Fields and Potential Functions
- Example proving a Vector Field is Conservative given a Potential Function

## Line Integrals

- Physical Understanding of a Line Integral
- Definition of a Line Integral
- Review Parameterization of Basic Curves
- Formula for Line Integral and Piecewise Smooth Curves
- Example #1 Evaluating a Line Integral along a circle
- Example #2 Evaluating a Line Integral along a Piecewise Smooth Curve
- Example #3 Evaluating a Line Integral along a line segment
- Example #4 Evaluating a Line Integral along a helix
- Formula for Evaluating a Line Integral with respect to x and y
- Example of Evaluating a Line Integral with respect to x and y
- Overview of Line Integrals of Vector Fields
- Example #1 Evaluating a Line Integral of a given Vector Field
- Example #2 Evaluating a Line Integral of a given Vector Field
- Example #3 Evaluating a Line Integral of a given Vector Field

## Review of Line Integrals

- Review and Formulas for Evaluating Line Integrals
- Example #1 Line Integral for a Piecewise Smooth Curve
- Example #2 Line Integral along a Curve
- Example #3 Line Integral for a Vector Field

## The Fundamental Theorem of Line Integrals

- Theorem for the Fundamental Theorem for Line Integrals
- Definitions and Facts about Line Integrals of Vector Fields
- Overview and Theorem of how to show a Vector Field is Conservative in 2-D
- Two Examples of showing a Vector Field is Conservative or not
- Overview and Example of how to find a Potential Function
- Example #1 Evaluating using the Fundamental Theorem for Line Integrals
- Example #2 Evaluating using the Fundamental Theorem for Line Integrals
- Overview of how to show a Vector Field is Conservative in 3-D
- Example of Evaluating using the Fundamental Theorem for Line Integrals in 3D

## Green’s Theorem

- Overview of Green’s Theorem and Positive Orientation
- Example #1 using Green’s Theorem
- Example #2 using Green’s Theorem
- Overview of how Green’s Theorem can be applied to regions with holes
- Example #3 using Green’s Theorem with polar coordinates
- Example #4 using Green’s Theorem with polar coordinates
- Application of Green’s Theorem for finding Area
- Example of Finding Area using Green’s Theorem in the reverse direction
- Example using Green’s Theorem to find work done by a force

## Curl and Divergence

- Overview of Curl and Divergence
- Basic Facts and Physical Interpretation of Curl and Divergence
- Example #1 calculating and evaluating the Curl and Divergence at a point
- Example #2 calculating and evaluating the Curl and Divergence at a point
- Example using the Curl to determine if a Vector Field is Conservative
- Example using Curl to show a Vector Field is Conservative and find a Potential Function
- Vector Forms of Green’s Theorem
- Example #1 of finding work done moving a body around a square
- Example #2 of finding area and work done moving an object about a curve

## Parametric Surfaces and Their Areas

- Overview of Parameterized Curves and Surfaces and Equations
- Example #1 Determining the Surface (plane) given a Parametric Representation
- Example #2 Determining the Surface (cone) given a Parametric Representation
- Example #3 Determining the Parametric Representation for an Elliptic Paraboloid
- Example #4 Determining the Parametric Surface for a Sphere
- Example #4 Determining the Parametric Surface for a Cylinder
- Overview of how to find the Tangent Plane to a Parametric Surface
- Example of how to write the Tangent Plane to a Parametric Surface
- Review of Surface Area and How to Find Surface Area of Parametric Surfaces
- Example #1 Find the Area of the Surface above a Triangle
- Example #2 Find the Surface Area for a Sphere inside a Cylinder

## Surface Integrals

- Overview of Surface Integrals for Scalar Functions
- Example #1 Evaluating a Surface Integral in the xy-plane
- Example #2 Evaluating a Surface Integral in the yz-plane
- Example #3 Evaluating a Surface Integral of a cylinder
- Example #4 Evaluating a Surface Integral of a paraboloid
- Overview of Normal Vectors and Surface Integrals for Vector Fields (Flux Integral)
- Example #1 Evaluating the Surface Integral for a Vector Field
- Example #2 Evaluating the Surface Integral for a Vector Field

## Stoke’s Theorem

- Overview of Stokes’ Theorem
- Relating Stokes’ Theorem to Line Integrals and Green’s Theorem and Surface Integrals
- Example #1 Verify Stokes’ Theorem
- Example #2 Verify Stokes’ Theorem
- Example #3 Evaluate using Stokes’ Theorem
- Example #4 Evaluate using Stokes’ Theorem

## The Divergence Theorem

- Overview of The Divergence Theorem
- Physical Interpretation of the Divergence Theorem
- Example #1 Evaluate using the Divergence Theorem for a surface box/a>
- Example #2 Evaluate using the Divergence Theorem for a triangular surface/a>
- Example #3 Evaluate using the Divergence Theorem for a circular cylinder
- Example #4 Evaluate using the Divergence Theorem for a sphere
- Example #5 Evaluate using the Divergence Theorem to find the flux