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))
d(Pn) = sqrt((P2-P1)^2 + (P3-P2)^2 + ... (Pn-P(n-1))^2)
Pythag essentially.
L = sqrt((x2-x1)^2 + (y2-y1)^2)
D = sqrt((z2-z1)^2 + L^2)
= sqrt((z^2-z1)^2+(x2-x1)^2+(y2-y1)^2)
Graphs of Simple Equations:
The graph of an equation in n variables is the set of all points in R^n that satisfy the equation. NOT all of the variables may appear in the equation!
Ex. x = 2 in R
-----------------|------------2--->
So it's a point.
x = 2 in R^2
Now it's a line in (x,y) (vertical line x=2).
x = 2 in R^3
Now it's a plane in (x,y) (can kind of think about it as moving/sliding).
(x-3)^2 + y^2 = 25 in R^2
Sqrt both sides.
The left side of the equation represents the distance between (3,0) and (x,y). The right side of the equation tells us this distance is always 5.
It's a circle.
(x-3)^2 + y^2 = 25 in R^3
It's a cylinder shifted in the z axis. This is because z can be any value, and the equation will still hold!
x + z = 10 in R^3
It's a line in R^2, thus a plane in R^3!
x^2+y^2+z^2 = 1 in R^3
It's a sphere!!! With its origin at (0,0,0) and every point being distant by 1.
Cylindrical and spherical coordinates:
Cylindrical ex.
(x,y,z) -> (r, theta, z)
Where r is = sqrt(x^2+y^2)
Where theta is = tan(y/x)
Spherical ex.
(x,y,z) -> (rou, theta, phi)
Where rou is = distance from origin (radius of the sphere)
Theta is still tan (y/x) (or, how much you have to rotate in the x,y plane)
Where phi is how much you have to go down from the z-axis.
Ex. Convert the point (-2, -2, 1) to cylindrical and spherical coordinates.
r = 5 -> cylinder beause any z value works.
tehta = pi/4 is the plane because it spins around the z axis
rou = 5 is the sphere because phi is the actual radius
phi = pi/4 is the cone because it is the amount of tilt from the z axis
r^2 = z is the smooth cone because z >=0.
rou = 4 sec phi because
Vectors:
A quantity that has both mangitued and direction.
Vectors can be represented by an arrow in R^n when n = 1, 2, 3.
The set of all vectors in R^n is dented by V^n.
Moving around a vector is still considered the same vector as long as its direction and magniude remains the same.
|u| is considered to be the lenght, or magnitude of vector u.
the first letter denotes the origin, and the ending letter denotes the terminal.