MATH314 Advanced Calculus

Triple Integral

Vector Calculus

Integral Theorems

Divergences theorem

Stokes theorems

Green theorem

Jacobians and change of variables

Cylindrical and spherical coordinates

Line integrals

Conservative fields

Scalar and vector fields

Application

Partial Derivatives

Limit laws

Continuity: limit at c is equal to f(c)

ε-δ definition of limit (2 variables)

Double Integral

Application

Surface Area

Polar Coordinate

Fubini's Theorem

Type I region

Type II region

Change order of integration

Used when x^2+y^2 appears in the integrand

x=rcos(θ), y=rsin(θ), dxdy= rdr*dθ

Moment: M = m*r

Moment of Inertia: I = m*r^2

Mass: m = integral[ ρ dA]

Joint PDF

Surface area of sphere: 4pi*r^2

General formula

Fubini's Theorem

3 Types of regions

Change order of intergration

Integrate to find the Volume

Clairaut's Theorem: f_xy = f_yx

Tangent Plane and Linear Approximation (切线的3D版)

Chain Rule

Directional Derivative

Implicit Function Theorem

Gradient Vector

Max directional derivative is norm of gradient

Gradient is normal to tangent surface

Local Max/Min/Saddle (when gradient=0)

Second derivative test (classify max or min or saddle)

Absolute Extrema

Domain should be a closed and bounded region

  1. Let gradient=0, find local extrema inside the domain
  1. Use Lagrange Multiplier to find extrema on the boundary

Use double integral to compute line integral

Use line integral to compute area

Independence of path: only endpoints matters

Closed curve integrates to 0

Scalar function

Vector field

Require: positively-oriented, simple, closed curve

Curl and Divergence

curl(F)=0 <=> F conservative

div(curl(F)) = 0 for all function F

Surface integrals

Integrate over scalar function

Integrate over vector fields (flux)