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UIC / Mathematics / MATH 210 / What do you mean by gradients?

What do you mean by gradients?

What do you mean by gradients?

Description

School: University of Illinois at Chicago
Department: Mathematics
Course: Calculus III
Professor: John steenbergen
Term: Fall 2016
Tags:
Cost: 25
Name: Math 210, Week 11 10-31 Notes
Description: These notes cover Steenbergen's notes from 10/31.
Uploaded: 11/07/2016
1 Pages 185 Views 0 Unlocks
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14.1 Vector FieldsDon't forget about the age old question of What makes describing even a simple concept more difficult than one might guess?

3 types of functions in this course

  1. r(t) = <x(t), y(t), z(t)>

One input, multiple outputs

  1. Z = f(x,y)         W = P(x,y,z)

Multiple inputs, one output

  1. Vector fields

        Multiple inputs, multiple outputsDon't forget about the age old question of Who was Gregor Mendel?

Vector fields:We also discuss several other topics like gsu human resources

20 = F(x,y) = <F1(x,y), F2(x,y)>

30 = G(x,y,z) = <g1(x,y,z), g2(x,y,z), g3(x,y,z)>We also discuss several other topics like Give an example how to solve a triangle.

Don't forget about the age old question of What are the common arrangements of backbone conformation?
Don't forget about the age old question of What is an alveolar air?

To draw a vector field, we draw at each point (a,b) the vector F(a,b)

Note: there are of course infinitely many parts (a,b) and vector F(a,b), so we can’t draw them all

Example: Draw F(x,y) = <1,1> (a constant vector field)

Example: Draw F(x,y) = <x,y> (“radial” vector field)

Example: Draw F(x,y) = <-y,x>

(1,0) → <0,1>

(0,1) → <-1,0>

(1,1) → <-1,1>

(1,1) → <-1,1>

(-2,0) → <0,2>

Important Note: Gradients are Vector Fields

Example: Z = X2 + Y2 (paraboloid surface)

Then, VF = <Fx,Fy> = <2x,2y> = VF ← vector field

Notation / Terminology: let F be a vector field (in 2d or 3d). If there is an function and such that F is its gradient, then we call it the “potential” function for F and F is called “conservative”

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