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Vector Addition
If u and v are vectors position so that the initial point of v is at the terminal point of u, then the sum u+v is the vector from the initial point of u to the terminal point of v

Scalar Multiplication
If c is a scalar and v is a vector, then the scalar multiple cv is the vector whose magnitude is c times the magnitude of v and whose direction is the same as v if c>0 and is opposite to v if c<0. If c=o or v=0, then cv=0.

Two nonzero vectors are parallel if
they are scalar multiples of one another

Normal Vector
Any vector that is perpendicular to the plane. The normal vectors of a plane containing vectors u and v are all the scalar multiples of uxv

For all ε>0, there exists δ>0 such that (x,y)∈Domainf and ⇒f(x,y)L<ε

A function f of two variables is continues at (a,b) if

 For D⊆R^{2}, we say f is continuous on D if f is continuous at every point (a,b)∈D

For all ε>0, there exists δ>0 such that if (x,y,z)∈Domainf and ⇒f(x,y,z)L<ε

For functions of n variables, define
Let x=<x _{1},x _{2},...,x _{n}> and a=<a _{1},a _{2},...,a _{n}> be vectors in R ^{n}. If f is defined on a subset D of R ^{n}, then means that for all ε>0, there exists δ>0 such that if x∈D and 0< x a<δ⇒f( x)L<ε

If z=f(x,y), then f is differentiable at (a,b) if
 Δz can be expressed in the form:
 Δz=f_{x}(a,b)(Δx)+f_{y}(a,b)(Δy)+ε_{1}Δx+ε_{2}Δy
 where ε_{1}→0 and ε_{2}→0 as (Δx,Δy)→(0,0)

Limit Definition of the Directional Derivative
 The Directional Derivative of f(x_{0},y_{0}) in the direction of a unit vector u=<a,b> is
 , provided the limit exists

Gradient Vector
If f(x,y), ∇f(x,y)=<f _{x}(x,y), f _{y}(x,y)>=

Local Maximum and Absolute Maximum
 f(a,b) is a local maximum value if f(x,y)≤f(a,b) for all points (x,y) in some disk with center (a,b)
 If this inequality holds for all points (x,y) in the domain of f then f has an absolute maximum at (a,b)

Local Minimum and Absolute Minimum
 f(a,b) is a local minimum if f(x,y)≥f(a,b) when (x,y) is near (a,b).
 If this inequality holds for all points (x,y) in the domain of f then f has an absolute minimum at (a,b)

Critical or Stationary Point
a point (a,b) is a critical point of f if f_{x}(a,b)=0 and f_{y}(a,b)=0, or if at least one of these partial derivatives does not exist.

Normal Line to a surface S in R^{3}
The line through P which is perpendicular to the tangent plane to S at P. The direction of the normal line is given by ∇F<x_{0},y_{0},z_{0}>

Double Integral of f over the rectangle R is
, if the limit exists

Triple integral of f over the box B
if the limit exists

Jacobian of the Transformation T given by x=g(u,v) and y=h(u,v) is

