MATH 241

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Vector & Space

Basic

Distance Formula in :

Equation of a Sphere :

unit vector:

Dot Product: , , angle between and is

Corollary:

Orthogonal:

Projection

Scalar projection of onto :

Vector projection of onto :

Cross Product: only in

The vector
, is the angle between two vectors.
Some properties of :
A volume of the parallelepiped determined by vectors , and , then

Lines

, is a vector parallels with the line, and is a vector to the line, is all vectors that point to the line.


represent the line concludes and , and when , the line is between and .

Planes

or , where is a normal vector, orthogonal to the plane, is just in the plane. so
skew: Not in the same plane. (异面)

Distances

For , in the space( ),

Vector Functions

Limits:

Continuity: vector function is continuous at a if

Differential

Derivatives:

Rules:

If , then is orthogonal to for all .

Integrals

Length:

Arc Length Function: ,

Curvature

Unit tangent vector:

Curvature: , , y=f(x):

Principal unit normal vector:

Binormal vector:

Normal Plane:

Osculating Plane:

Torsion: ,

Partial Derivatives

Limits and Continuity

Definitio of Continuity: A function of two variables is called continuous at if .

Derivatives

Chain Rule:

Directional Derivatives

Definitio: for

Gradient: ,

Maximum and Minimum Values

Local Max/Min: has a local max/min/saddle point at and the first-order partial derivatives of f exist there, then

Second Derivatives Test: For , Let ,

Lagrange Multipliers

Find the maximum and minimum values of with , we:

Find all values of such that: , then calculate.

If there are two constraints, use

Multiple Integrals

Double Integral

If is continuous on the rectangle , then

Some other type of region D:

Average Value:

Polar: Polar rectangle D given by where

Surface Area:

Vector Calculus

Line Integrals

Definitio: