Math Insight

Thread: Multivariable calculus

Covers basic pages in multivariable calculus

1. Differential Calculus
   1. Partial derivatives
      • Introduction to partial derivatives
      • Partial derivative examples
      • Partial derivative by limit definition
   2. Differentiability and the derivative
      • Introduction to differentiability in higher dimensions
      • The definition of differentiability in higher dimensions
      • Subtleties of differentiability in higher dimensions
      • The multivariable linear approximation
      • The derivative matrix
      • Examples of calculating the derivative
      • The multidimensional differentiability theorem
      • Non-differentiable functions must have discontinuous partial derivatives
      • A differentiable function with discontinuous partial derivatives
   3. The chain rule
      • Introduction to the multivariable chain rule
      • Special cases of the multivariable chain rule
      • Multivariable chain rule examples
   4. The directional derivative and the gradient
      • An introduction to the directional derivative and the gradient
      • Derivation of the directional derivative and the gradient
      • Directional derivative and gradient examples
   5. Applications of differential calculus
      • Introduction to Taylor's theorem for multivariable functions
      • Multivariable Taylor polynomial example
      • Introduction to local extrema of functions of two variables
      • Two variable local extrema examples
2. Integral calculus
   1. Double integrals
      • Introduction to double integrals
      • Double integrals as iterated integrals
      • Double integral examples
      • Double integrals as volume
      • Double integrals as area
      • Examples of changing the order of integration in double integrals
      • Double integrals where one integration order is easier
   2. Triple Integrals
      • Introduction to triple integrals
      • The shadow method for determining triple integral bounds
      • The cross section method for determining triple integral bounds
      • Triple integral examples
   3. Changing variables
      • Introduction to changing variables in double integrals
      • Area calculation for changing variables in double integrals
      • Double integral change of variable examples
      • Illustrated example of changing variables in double integrals
      • Triple integral change of variables story
      • Introduction to changing variables in triple integrals
      • Volume calculation for changing variables in triple integrals
      • Triple integral change of variable examples
3. Curves and surfaces
   1. Parametrized curves
      • An introduction to parametrized curves
      • Derivatives of parameterized curves
      • Parametrized curve and derivative as location and velocity
      • Tangent lines to parametrized curves
      • Tangent line to parametrized curve examples
      • Orienting curves
   2. Length of curves
      • The arc length of a parametrized curve
      • Parametrized curve arc length examples
   3. Parametrized surfaces
      • An introduction to parametrized surfaces
      • Parametrized surface examples
      • Normal vector of parametrized surfaces
      • Orienting surfaces
      • A Möbius strip is not orientable
   4. Surface area
      • Surface area of parametrized surfaces
      • Calculation of the surface area of a parametrized surface
      • Parametrized surface area example
4. Vector fields
   1. Vector field basics
      • Vector field overview
      • Vector fields as fluid flow
   2. Vector operators
      • The idea of the divergence of a vector field
      • Subtleties about divergence
      • The idea of the curl of a vector field
      • Subtleties about curl
      • The components of the curl
      • Divergence and curl notation
      • Divergence and curl example
      • The definition of curl from line integrals
      • Calculating the formula for circulation per unit area
5. Integration over curves and surfaces
   1. Line integrals
      • Introduction to a line integral of a scalar-valued function
      • Introduction to a line integral of a vector field
      • Line integrals are independent of parametrization
      • Examples of scalar line integrals
      • Alternate notation for vector line integrals
      • Vector line integral examples
      • Line integrals as circulation
   2. Surface integrals
      • Introduction to a surface integral of a scalar-valued function
      • Introduction to a surface integral of a vector field
      • Scalar surface integral examples
      • Vector surface integral examples
   3. Integration Synopsis
      • The integrals of multivariable calculus
      • Length, area, and volume factors
6. The fundamental theorems of vector calculus
   1. Gradient theorem for line integrals
      • An introduction to conservative vector fields
      • The gradient theorem for line integrals
      • A simple example of using the gradient theorem
      • How to determine if a vector field is conservative
      • A conservative vector field has no circulation
      • A path-dependent vector field with zero curl
      • Testing if three-dimensional vector fields are conservative
      • Finding a potential function for conservative vector fields
      • Finding a potential function for three-dimensional conservative vector fields
   2. Green's theorem
      • The idea behind Green's theorem
      • When Green's theorem applies
      • Other ways of writing Green's theorem
      • Green's theorem examples
      • Using Green's theorem to find area
      • Green's theorem with multiple boundary components
   3. Stokes' theorem
      • The idea behind Stokes' theorem
      • Proper orientation for Stokes' theorem
      • Stokes' theorem examples
   4. Divergence theorem
      • The idea behind the divergence theorem
      • Divergence theorem examples
   5. Fundamental theorem synopsis
      • The fundamental theorems of vector calculus

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