Calculus Notes/ReviewYoung Tai Ahn

Calculus

The Fundamental Theorem of Calculus baF (x)dx = F (b) F (a) where F is continuous on [a, b].

Vector Fields

Definitions vector field, gradient vector field, conservative vector field, potential function

Vector Field Let O be a subset of Rn. A vector field on Rn is a function F that assigns to each point u O a n-dimensionalvector F (u).

Gradient Given a function f on Rn, the gradient of f , Of = f1, f2, fn.Conservative Vector Field A vector field F is a conservative vector field if it is the gradient of some function f . So

F = Of . Here, f is called a potential function for F .

Line Integrals

Definitions line integral, arc length, line of integral of f on C with respect to x, y, line integral with respect to arc length,orientation of C,

Line Integral Given f defined on a smooth curve in C R2, C given as x = x(t), y = y(t), a t b, the line integral off along C is

Cf(x, y)ds.

Formula for Line Integral If f is continuous, thenCf(x, y)ds =

baf(x(t), y(t))

(dxdt )

2 + (dydt )2dt.

Note Line integrals work in R3 (maybe Rn) and in vector fields.

The Fundamental Theorem for Line Integrals

The Theorem Let C be a smooth curve given by the vector function r(t), a t b. Let f be a function in R2, R3 whosegradient vector Of is continuous on C. Then

COf d(r) = f(r(b)) f(r(a)).

Definitions Idnependence of path, closed curve, open domain, connected domain, simple curve, simply-connected region

Theorem A conservative vector field

Greens Theorem

Definitions positive/negative orientation,

Greens Theorem

Curl and Divergence

Definitions Curl, Divergence, Laplace operator