Vector Calculus (Decoded)
A Streamlined Approach

Conquer complex concepts with ease — Gain a solid understanding of Vector Calculus applications — Tackle advanced problems confidently

Vector Field

32 min 6 Examples

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 Integral

1 hr 37 min 9 Examples

Physical Understanding of a Line Integral
Defintion 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

23 min 3 Examples

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

Fundamental Theorem Of Line Integrals

40 min 5 Examples

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

1 hr 6 Examples

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

1 hr 1 min 6 Examples

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 Surface

1 hr 6 min 8 Examples

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 #5 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 Integral

1 hr 28 min 6 Examples

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

Stokes’ Theorem

1 hr 12 min 4 Examples

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

Divergence Theorem

57 min 5 Examples

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

Vector Calculus Review

6 min 1 Examples

Recap of Vector Calculus

Chapter Test

1 hr 37 min 14 Practice Problems

Determine whether the statement is true or false (Problems #1-3)
Determine whether F is conservative and, if so, find the potential function (Problem #4)
Calculate the curl and divergence at the given point (Problem #5)
Use the Fundamental Theorem of Line Integrals to evaluate (Problem #6)
Use Green’s Theorem to evaluate (Problem #7)
Find the Work done by a force moving a particle along a straight line (Problem #8)
Sketch the Vector Field and Sketch the Gradient Field (Problems #9a and 9b)
Evaluate the Line Integral (Problem #10)
Find the equation of the tangent plane to the given parametric surface (Problem #11)
Evaluate the Surface Integral (Problem #12)
Use Stoke’s Theorem to evaluate the integral (Problem #13)
Use the Divergence Theorem to calculate the surface integral (Problem #14)