Calculus IIICalculus III

Arrows with length and direction: how to add them, scale them, and multiply them with the dot and cross products.

1. 1Vectors: The Basics
2. 2Vector Arithmetic
3. 3The Dot Product
4. 4The Cross Product

Lines, planes and surfaces in 3D, functions of several variables, curves traced by vector functions, and the cylindrical and spherical coordinate systems.

1. 5The 3-D Coordinate System
2. 6Equations of Lines
3. 7Equations of Planes
4. 8Quadric Surfaces
5. 9Functions of Several Variables
6. 10Vector Functions
7. 11Calculus with Vector Functions
8. 12Tangent, Normal and Binormal Vectors
9. 13Arc Length with Vector Functions
10. 14Curvature
11. 15Velocity and Acceleration
12. 16Cylindrical Coordinates
13. 17Spherical Coordinates

Rates of change for functions of several variables: limits in the plane, partial derivatives, the multivariable chain rule, and the gradient.

1. 18Limits of Functions of Several Variables
2. 19Partial Derivatives
3. 20Interpreting Partial Derivatives
4. 21Higher Order Partial Derivatives
5. 22Differentials
6. 23The Chain Rule
7. 24Directional Derivatives and the Gradient

Tangent planes and linear approximation, critical points of surfaces, absolute extrema on regions, and optimization with constraints.

1. 25Tangent Planes and Linear Approximations
2. 26The Gradient, Tangent Planes and Normal Lines
3. 27Relative Minimums and Maximums
4. 28Absolute Minimums and Maximums
5. 29Lagrange Multipliers

Integrating over regions and solids: iterated integrals, polar, cylindrical and spherical coordinates, the Jacobian, and surface area.

1. 30Double Integrals
2. 31Iterated Integrals
3. 32Double Integrals over General Regions
4. 33Double Integrals in Polar Coordinates
5. 34Triple Integrals
6. 35Triple Integrals in Cylindrical Coordinates
7. 36Triple Integrals in Spherical Coordinates
8. 37Change of Variables
9. 38Surface Area

Vector fields, integrating along curves, the work done by a force, conservative fields, and Green's theorem.

1. 39Vector Fields
2. 40Line Integrals of Functions
3. 41Line Integrals of Vector Fields
4. 42The Fundamental Theorem for Line Integrals
5. 43Conservative Vector Fields
6. 44Green's Theorem

Curl and divergence, parametric surfaces, flux through a surface, and the theorems of Stokes and Gauss that tie it all together.

1. 45Curl and Divergence
2. 46Parametric Surfaces
3. 47Surface Integrals
4. 48Flux Integrals
5. 49Stokes' Theorem
6. 50The Divergence Theorem