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MATH314 Advanced Calculus - Coggle Diagram
MATH314 Advanced Calculus
Triple Integral
Jacobians and change of variables
Cylindrical and spherical coordinates
Fubini's Theorem
3 Types of regions
Change order of intergration
Integrate to find the Volume
Vector Calculus
Line integrals
Scalar function
Vector field
Conservative fields
Independence of path: only endpoints matters
Closed curve integrates to 0
Scalar and vector fields
Surface integrals
Integrate over scalar function
Integrate over vector fields (flux)
Integral Theorems
Divergences theorem
Stokes theorems
Green theorem
Use double integral to compute line integral
Use line integral to compute area
Require: positively-oriented, simple, closed curve
Application
Curl and Divergence
curl(F)=0 <=> F conservative
div(curl(F)) = 0 for all function F
Partial Derivatives
Limit laws
Continuity: limit at c is equal to f(c)
ε-δ definition of limit (2 variables)
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
Let gradient=0, find local extrema inside the domain
Use Lagrange Multiplier to find extrema on the boundary
Double Integral
Application
Moment: M = m*r
Moment of Inertia: I = m*r^2
Mass: m = integral[ ρ dA]
Joint PDF
Surface Area
Surface area of sphere: 4pi*r^2
General formula
Polar Coordinate
Used when x^2+y^2 appears in the integrand
x=r
cos(θ), y=r
sin(θ), dx
dy= r
dr*dθ
Fubini's Theorem
Type I region
Type II region
Change order of integration