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# Calculus 3 with the Math Sorcerer

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Video Introducing this tutorial

Vectors in the Plane :
Introduction to Vectors in the Plane
How to Write a Vector in Component Form and Sketch in the Plane
Find the Vectors Given the Initial and Terminal Points Example 1
Find the Vectors Given the Initial and Terminal Points Example 2
Unit Vectors and Standard Unit Vectors
Find the Magnitude Example 1
Find the Magnitude Example 2
Find the Terminal Point Given the Vector and the Initial Point
Find a Unit Vector in the Direction of Another Vector Example 1
Find a Unit Vector in the Direction of Another Vector
Vector in the Direction of Another Vector with a Given Magnitude Example 1
Vector in the Direction of Another Vector with a Given Magnitude Example 2

Space Coordinates and Vectors in Space :
Introduction to Space Coordinates
Find the Magnitude of the Vector in Space Example 1
Find the Magnitude of the Vector in Space Example 2
Find the Magnitude of the Vector in Space Example 3
Vector in Space in the Direction of Another Vector with a Given Magnitude Ex 1
Vector in Space in the Direction of Another Vector with a Given Magnitude Ex 2
Find a Unit Vector in the Direction of v and Opposite of v
The Dot Product
Introduction to the Dot Product
Dot Product of Vectors in the Plane Example 1
Dot Product of Vectors in the Plane Example 2
Dot Product of Vectors in Space Example 1
Dot Product of Vectors in Space Example 2
Determine if the Three Dimensional Vectors are Orthogonal
Find the Angle Between the Vectors in the Plane
Find the Angle Between the Vectors in Space Example 1
Find the Angle Between the Vectors in Space Example 2
Dot Product Given the Magnitudes and the Angle Between the Vectors Example 1
Dot Product Given the Magnitudes and the Angle Between the Vectors Example 2
Find Two Vectors in Opposite Directions Orthogonal to Another Vector

Projections :
Introduction to Projections
Projections Example 1
Projections Example 2

Direction Cosines and Direction Angles :
Introduction to Direction Cosines and Direction Angles
Example 1
Example 2
Example 3

The Cross Product :
Find the Cross Product Example 1
Find the Cross Product Example 2
Find the Cross Product Example 3
Find a Unit Vector Orthogonal to Two Vectors in Space Example 1
Find a Unit Vector Orthogonal to Two Vectors in Space Example 2

Distance in Space :
Find the Distance Between Two Points in Space Example 1
Find the Distance Between Two Points in Space Example 2

The Sphere :
Find the Equation of a Sphere Given the Center and Radius Example 1
Find the Equation of a Sphere Given the Center and Radius Example 2
Find the Equation of the Sphere Given the Endpoints of the Diameter
Write the Equation of the Sphere in Standard Form by Completing the Square

Lines in Space :
The Formulas for the Symmetric and Parametric Equations of a Line in Space
Find the Equation of the Line Given a Point and Parallel Vector Example 1
Find the Equation of the Line Given a Point and Parallel Vector Example 2
Find the Equation of the Line Given a Point and Parallel Vector Example 3
Find the Equation of the Line Given a Point and a Parallel Line
Find the Equation of a Line Passing through Two Points

Planes in Space :
Derivation of the Equation of the Plane
Find the Equation of Plane Given a Point and a Normal Vector Example 1
Find the Equation of the Plane Given a Point and a Normal Vector Example 2
Find the Equation of a Plane Given Three Points
Find the Equation of the Plane that is Parallel to the z-axis Given Two Points
Find the Equation of the Plane Given a Point and a Perpendicular Line
Find the Equation of the Plane Given a Point and a Line on the Plane
Find the Equation of the Plane Given Two Points and a Perpendicular Plane

Cylindrical Coordinates :
Introduction to Cylindrical Coordinates
Graphs in Cylindrical Coordinates
How to Convert Rectangular Equations to Cylindrical Equations
Converting Rectangular Coordinates to Cylindrical Coordinates
Convert Rectangular to Cylindrical Example
Converting Cylindrical Coordinates to Rectangular Coordinates
Convert Cylindrical to Rectangular Example 1
Convert Cylindrical to Rectangular Example 2
Convert Cylindrical to Rectangular Example 3
Convert Cylindrical to Rectangular Example 4

Spherical Coordinates :
Introduction to Spherical Coordinates
Convert the Spherical Coordinates to Rectangular Coordinates Example 1
Convert the Spherical Coordinates to Rectangular Coordinates Example 2
Convert the Spherical Coordinates to Rectangular Coordinates Example 3
Convert Rectangular Coordinates to Spherical Coordinates Example 1
Convert Rectangular Coordinates to Spherical Coordinates Example 2
How to Convert Spherical Equations to Rectangular Equations
How to Convert Rectangular Equations to Spherical Equations

Vector-Valued Functions :
Introduction to Vector-Valued Functions
Graph of the Vector-Valued Function, Orientation, and Rectangular Form Example 1
Graph of the Vector-Valued Function, Orientation, and Rectangular Form Example 2
How to Represent Plane Curves as Vector Valued Functions (Multiple Examples)
How to Represent Any Circle as a Vector Valued Function (Mutiple Examples)
How to Represent Any Circle as a Vector Valued Function (More Examples)
How to Represent Any Ellipse as a Vector Valued Function (Multiple Examples)
How to Represent Any Hyperbola as a Vector Valued Function (Multiple Examples)
Represent the Plane Curve as a Vector Valued Function (Circle Example)
Represent the Plane Curve as a Vector Valued Function (Parabola Example)
Represent the Plane Curve as a Vector Valued Function (Line Example)

Limits of Vector-Valued Functions :
Find the Limit Example 1
Find the Limit Example 2
Find the Limit Example 3
Find the Limit Example 4
Intervals on Which Vector-Valued Function is Continuous Example 1
Intervals on Which Vector-Valued Function is Continuous 2

Derivatives of Vector-Valued Functions :
Derivative Example 1
Derivative Example 2
Derivative Example 3
Derivative Example 4
Derivative Example 5
Derivative Example 7
Derivative Example 8
Derivative Example 9
Derivative Example 10
Derivative Example 11
Second Derivative Example 1
Second Derivative Example 2
Intervals where Vector-Valued Function is Smooth
Intervals where Vector-Valued Function is Smooth Example 2
Differential Equation with Vector Valued Function

Integrals of Vector Valued Functions :
Indefinite Integral Example 1
Indefinite Integral Example 2
Indefinite Integral Example 3
Indefinite Integral Example 4
Definite Integral Example

Velocity, Speed, and Acceleration :
Velocity, Speed, and Acceleration Given a Vector-Valued Function Example 1
Velocity, Speed, and Acceleration Given a Vector-Valued Function Example 2
Velocity, Speed, and Acceleration Given a Vector-Valued Function Example 3
Velocity, Speed, and Acceleration Given a Vector-Valued Function Example 4
Find the Acceleration Given the Speed and Position Functions Example 1
Find the Acceleration Given the Speed and Position Functions Example 2

Tangent Vectors and Normal Vectors :
Introduction to the Unit Tangent Vector
Unit Tangent Vector Example 1
Unit Tangent Vector Example 2
Unit Tangent Vector Example 3
Unit Tangent Vector Example 4
How to Find the Equation of a Tangent Line in Space Example 1
How to Find the Equation of a Tangent Line in Space Example 2
Introduction to the Principal Unit Normal Vector
How to Find the Principal Unit Normal Vector
How to Find the Principal Unit Normal Vector (Harder Example)

Arc Length and Curvature :
Introduction to Arc Length
The Arc Length Function
Introduction to Curvature
The Curvature of a Circle
How to Compute Curvature
Curvature Example 1
Curvature Example 2
Curvature Example 3

Functions of Several Variables :
Introduction to Functions of Several Variables
Domain and Range of Multivariable Functions
Evaluate the Multivariable Function
Find the Domain and Range Example 1
Find the Domain and Range Example 2
Find the Domain and Range Example 3
How to Find the Level Curves of a Function
Level Curves Example 1
Level Curves Example 2
Level Curves Example 3
Level Curves Example 4
Level Curves Example 5

Limits and Continuity :
Find the Limit Example 1
Find the Limit Example 2
Find the Limit Example 3
Find the Limit Example 4
Limit using Polar Coordinates Example 1
Limit using Polar Coordinates Example 2
Limit using Polar Coordinates Example 3
Limit using Polar Coordinates Example 4
How to Show a Limit Does Not Exist

Partial Derivatives :
Introduction to Partial Derivatives
More Examples of Partial Derivatives
Partial Derivative Example 1
Partial Derivative Example 2
Partial Derivative Example 3
Partial Derivative Example 4
Partial Derivative Example 5
Partial Derivative Example 6
Partial Derivative Example 7
Introduction to Higher Order Partial Derivatives HQ
Find All Second Order Partial Derivatives HQ

Laplace's Equation, Heat Equation, Wave Equation, and Cauchy Riemann Equations :
Laplace's Equation
The Heat Equation
The Wave Equation
The Cauchy Riemann Equations

Differentials :
Introduction to the Total Differential
Total Differential Example 1
Total Differential Example 2
Total Differential Example 3
Total Differential Example 4
Total Differential Example 5
Total Differential Example 6
Using the Total Differential to Make an Approximation

The Chain Rules :
Introduction to the Chain Rule for One Independent Variable
Chain Rule for One Independent Variable Example
Introduction to the Chain Rule for Two Independent Variables
Chain Rule Example 1
Chain Rule Example 2
Chain Rule Example 3
Chain Rule Example 4

Implicit Differentiation :
Finding dy/dx using Calculus 3
Multivariable Version of Implicit Differentiation
Example 1
Example 2
Example 3

Introduction to Directional Derivatives
What is the Gradient Vector and What Does it Mean?
Introduction to the Gradient Theory and Formulas
Directional Derivative Example 1
Directional Derivative Example 2
Directional Derivative Example 3
Directional Derivative Example 4
Directional Derivative Example 5
Directional Derivative Example 6
Find a Normal Vector to the Plane
Find a Unit Normal Vector to the Plane
Find a Unit Normal Vector to the Sphere
Find the Gradient and Maximum Value of the Directional Derivative Example 1
Find the Gradient and Maximum Value of the Directional Derivative Example 2

Tangent Planes and Normal Lines :
Introduction to Tangent Planes and Normal Lines
Example 1
Example 2
Example 3

Extrema of Functions of Two Variables :
The Second Partials Test
How to Use the Second Partials Test to Find Relative Extrema Example 1
How to Use the Second Partials Test to Find Relative Extrema Example 2

Applications of Extrema :
Minimum Distance from a Plane to a Point

Lagrange Multipliers :
How to Use Lagrange Multipliers
Example 1
Example 2
Example 3
Example 4
Example 4
Example 5
Lagrange Multipliers with Two Constraints

Iterated Integrals and Area in the Plane :
Integrating Multivariable Functions with Respect to One Variable
Example of Integrating with Respect to One Variable
Introduction to Iterated Integrals
Switching the Order of Integration Example 1
Switching the Order of Integration Example 2
Switching the Order of Integration Example 3
Switching the Order of Integration Example 4
Switch the Order Integration and Find the Area
Sketch the Region Given by the Iterated Integral and Find the Area
Evaluate the Iterated Integral Example 1
Evaluate Iterated Integral Example 2
Iterated Integral Example 3
Iterated Integral Example 4

Double Integrals and Volume :
Introduction to Double Integrals and Volume
How to Write Both Orders of Integration Example 1
Write Both Orders of Integration Example 2
Write Both Orders of Integration Example 3

Triple Integrals :
Introduction to Triple Integrals
Evaluate the Triple Integral

Vector Fields :
Introduction to Vector Fields
Conservative Vector Field and Potential Function Example 1
Conservative Vector Field and Potential Function Example 2
Conservative Vector Field and Potential Function Example 3
Determine if the Vector Field is Conservative Example 1
Determine if the Vector Field is Conservative Example 2
Determine if the Vector Field is Conservative Example 3
Determine if the Vector Field is Conservative Example 4
Divergence Example 1
Divergence Example 2
Divergence Example 3
Divergence Example 4
Divergence Example 5
Curl Example 1

Line Integrals :
Introduction to Line Integrals
Line Integral Example 1
Line Integral Example 2
Line Integral Example 3
Line Integral Example 4
Line Integral Example 5
Line Integral of a Vector Field Example 1
Line Integral of Vector Field Example 2
Line Integral of Vector Field Example 3
Line Integral in Differential Form Example 1
Line Integral in Differential Form Example 2
Work Done by the Force Field