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.
Multiplication by a scalar:
Given scalar c
If c > 0, cu is pointing in the same direction as u.
If c < 0, cu is pointing in the opposite direction as u.
Magnitude is always positive, thus = |c|*||u||
----> u
c > 0
--------------> cu
c < 0
<---- cu
Unit vector:
A vector that has a magnitude of 1. Given by:
u = v/||v||
Addition of vectors:
Their sum is formed by using the tail-to-tip method.
u + v = v + u
u - v
can be found by doing u + (-v)
Standard basis vectors
Vectors that point in the direction of the coordinate axis, and are unit vectors.
Otherwise known as i, j, k, etc.
They are referred to as e1, e2...en.
In V^2, we call i, j
Where x = i, and y = j.
In V^3, we call i, j, k
They enable us to represent any vector in V^n via:
v = ae + a2e2 + ... + anen
Which we may often also write as:
v = <a1, a2, ... an>
Which is known as component form, as a1, a2, ... an are known as the components.
Given point P (a1, a2, ... an), its position vector OP = <a1, a2, ..., an>
Given point A (a1, a2, ... an) and B (b1, b2, ... bn)
AB = <b1 - a1, b2 - a2, ... bn - an>
Given v = <a1, a2, ... an>, ||v|| = sqrt(a1^2 + a2^2 + ... + an^2)
Vector properties:
a + b = b + a
a + (b + c) = (a + b) + c
a + 0 = a
a + (-a) = 0
r(a + b) = ra + rb
(r+s)a = ra + sa
(rs)a = r(sa)
1a = a, (-1)a = -a
Dot Products:
(a*b)c DOES NOT NECESSARILY = a(b*c)
Because they can be totally different directions!
a*a = |a|^2
Geometric interpretation of the dot product:
Magnitudes have to be positive.
cos(theta) = a*b/|a||b| can be proven using Law of Cosines + |a|^2 = a * a
Do proof at home!
if a*b = 0, then a and b are orthogonal.
Because a*b = |a||b|*cos and |a||b| is positive,
the sign of cos(theta) = the sign of a*b.
Thus, a*b > 0 a and b are acute
If a*b < 0, then a and b are obtuse.
Because cos(theta) <= 1, |a*b| < |a||b|
Cross Product:
axb is orth. to a AND b
The direction of a x b is given by the right hand rule.
|a x b| = |a||b| sin(theta)
Yields a vector.
Square matrix = equal number of columns and height.
Determinants are calculated using square matrixies.
The area of the parallelogram formed by two vectors is given by the magnitude of the cross product of the two vectors,
A = ||a x b||