Multivariable Calc Theorems

Front 1

Linear Dependence

Back 1

A set in V is called linearly dependent if there is a finite set of elements in S, a set a scalars, not all zero, such that the sum of cx is zero.

Front 2

Linear Independence

Back 2

A set in V is called linearly independent if the sum of cx implies that all constants c are equal to zero.

Also, independent if and only if det does not equal 0.

Front 3

Basis

Back 3

A finite set of elements in V the is independent and spans V.

Front 4

Dimension

Back 4

Linear space V has a basis of n elements, the integer n is called the dimension of V.

Front 5

Open Set

Back 5

A set A in Rn is called open if all its points are interior points.

Front 6

Differentiable Scalar Field

Back 6

We say that f is differentiable at a id there exists a linear transformation Ta:Rn--->R from Rn to R, and a scalar function E(a,v) such that f(a+v) = f(a) + Ta(v) + |v|E(a,v), there E(a,v)---->0 as |V|--->0. Ta is the total derivative.

Front 7

Chain Rule (scalar fields)

Back 7

Let f be a scalar field defined on open set S in Rn, and let r be a vector function which maps an interval J from R1 into S. Define the composite g=for on J by:
g(t) = f(r(t)) if t is in J.
Let t be a point at which r'(t) exists and assume f is diff. at r(t). Then g'(t) exists and is equal to grad(f(r(t)))*r'(t).

Front 8

Chain Rule (vector fields)

Back 8

Let f anf g be vector fields such that the composition is deinfed in the neighborhood of a point A. Assume that g is differentiable at a, with total derivative g'(a). Let b=g(a) and assume f differentiable at b, with total deriv. at f'(b). Then total deriv is given by:
h'(a)=f'(b)og'(a).

Front 9

Sufficient Condition for the Equality of Mixed Partials

Back 9

Let f be a scalar field with all partials that exists on open set S. If (a,b) is a point at which both D12 and D21 are continuous, then D21=D12 at that point.

Front 10

Smooth

Back 10

J=[a,b] be a finite closed interval in R1. The path is smooth if the derviative exists and is continuous in the open interval (a,b). The path is called peicewise smooth is the interval [a,b] can partitioned into a finite number of subintervals in each of which the path is smooth.

Front 11

Connectedness

Back 11

The S is connected if every pair of points in S can be joined by a piecewise smooth path whose path lies in S.

Front 12

Second Fundamental Theorem of Calculus

Back 12

Let P be a differentiable scalar field with a continuous gradient grad(P) on an open connected set S in Rn. Then for any two points a and b joined by a peicewise smooth path in S we have:

integral from a to b of grad(P)*ds = P(b)-P(a).

Front 13

First Fundamental Theorem of Calculus

Back 13

Let f be a vector field that is continuous on an open connected set S in Rn, and assume that the line integrals of f is independent of the path in S. Let a be a fixed point of A and define a scalar field P on S by the equation
p(x) = integral from a to x of f*dr where r is any piecewise smooth path in S joining a to x. Then the gradient exists and is equal to f; that is, grad(P)=f(x) for all x in S.

Front 14

Necessary and Sufficient Conditions for a Vector Field to be a Gradient

Back 14

f is a vector field continuous on open connected set S in Rn. Then the 3 statements are equal:
-f is the gradient of some potential function.
-The line integral of f is independent of path in S.
-The line integral of f is zero around every piecewise smooth closed path.

Front 15

Step Function

Back 15

A function f defined on a rectangle Q is said to be a step function if a partition P of Q exists such that f is constant on each of the open subrectangles of P.

Front 16

Integral of a Bounded Function over a Rectangle

Back 16

If there is one and only one number I such that double int of S =< I =< double int of T for every pair of step functions satifying these inequalities, this number I is called the double integral of f over Q.

Front 17

Integrability of Continuous Fns

Back 17

If a function f is continuous on a rectangle Q = [a,b]x[c,d], then f is integrable on Q.

Front 18

Bounded Set of Content Zero

Back 18

A is a bounded subset of a plane. A has content zero is for every E>0 there is a finite set a rectangle whose union contains A and the sum of whose areas does not exceed epsilon.

Front 19

Green's Theorem

Back 19

Let P and Q be scalar fields that are continuously differentiable on an open set S in the xy-plane. Let C be a piecewise smooth Jordan curve, and let R denote the union of C and its interior. Assume R is a subset of S. Then we have: (already know it).

Front 20

Stokes' Theorem

Back 20

-S is smooth simple parametric surface, T is a region in the uv-plane bounded by a peicewise smooth Jordan curve.
-Assume r is one-to-one.
-r is in C2 on an open set.
-P Q R continuously differentiable scalar fields on S.

Front 21

Condition for Gradient Fields

Back 21

Let F be cont. diff. vector field on open convex set in R3. Then F is a gradient on S iff curlF = 0 on S.

Front 22

Divergence Theorem

Back 22

-Solid bounded by an orientable closed surface A.
-n= unit outer normal
-F in continuously differentiable.