Visualizing 3D Objects

Single-variable calculus:
f : D c R -> R

D = domain

c = subset

R = real numbers

f = function

Ex. f(x) = x/1-x

where x-> x/1-x

R^a where a represents the number of dimensions.

Multivariable calculus:
f : D c R^n -> R

Basically, for something like R^2, we would input two (x,y).

The graphs themselves exist in R^n+1 (as we need one additional dimension to represent them).

So, for R^2, the graph would live in R^3, a three dimensional space.

Ex. f(x,y) = (x+y)^2

R^2 -> R

(x,y) "is assigned to" (x+y)^2

n = 2, Dimension = 3

f(x,y,z) = x^2+y^2+z^2

R^3 -> R

(x,y,z,) "is assigned to" x^2+y^2+z^2

n = 3, Dimension = 4

Vector valued functions:

r : D c R^k-R^n

where r is the vector valued function.

k = 1 or 2

assigns numbers in R^n

Ex. If k = 1

r(t) = <cos(t), sin(t), t> <--- a curve

R^1 -> R^3

If k = 2
r(u,v,) = <u, v, (u+v)^2>

R^2 -> R^3 <--- a surface

Vector fields

F : D c R^n -> R^n

Ex. if n = 2

F(x,y) = <x,1>

if n = 3

F(x,y,z) = <cos(x), y*sin(x), z^2 + y>

Three Dimensional Space

The right hand rule!

Index points towards x, middle y, thumb z.

Conventional layout:
y-axis on the right

x-axis coming towards

z-axis going upwards.

Graphing:

Distance Formula:

d(P1,P0) = sqrt((x1-x0)^2 + (y1-y0)^2)

d(P1,P2) = sqrt((x2-x1)^2 + (y2-y1)^2 + (z2-z1)^2))