1 The Dot Product

The dot product is defined as:

A ·B = |A||B| cos θ

This can be used to find the angle betweentwo lines. Two vectors are perpendicular iftheir dot product is 0.

2 The Cross Product

The cross product is an operation that pro-duces a vector perpendicular to two other vec-tors. Additionally, it is the area of the parallel-ogram formed by the two vectors it is used on.

a× b = |a||b|n sin θ

The volume of a parallelopiped composed ofvectors a,b, and c is:

|(a× b) · c|

3 Planes

The definition of a plane through vectorr0 =< x0, y0, z0 > with normal n is:

n · (r− r0) = 0

Two planes are parallel if their normals areequal. The angle between two planes is definedas the angle between their normals. The anglebetween a line (b) and a plane with normaln can be found using:

n · b = |n||b| sin θ

4 Projection

A projection is when a line is ”dropped” onanother. The projection of vector w on v canbe found using:

Projw v =v ·w|w||w|w

5 Quadratic Surfaces

A quadratic surface takes the formax2 + by2 + cz2 = 1. If a,b, and c arepositive, the surface is an ellipsoid. If onecoefficient is negative, it’s a hyperboloid ofone sheet. If two terms are negative, it’s ahyperboloid of two sheets. If the constant atthe end (1 in this case) is 0, it’s a cone. If oneterm is missing, it’s a cylinder. If one term ismissing a degree and there is one negative, it’san elliptic paraboloid. In the same case withtwo negatives, it’s a hyperbolic paraboloid.

6 Arc Length for Vector Functions

Given the vector r(t) =< f1(t), f2(t), f3(t) >the arclength is:∫ b

a

√f ′1(t)

2 + f ′2(t)2 + f ′3(t)

2 dt

7 Polar Coordinates

x = r cos θ, y = r sin θ, z = z

r =√x2 + y2, θ = arctan y

xThe Jacobian is r.

8 Spherical Coordinates

x = ρ sinφ cos θ, y = ρ sinφ sin θ, z = ρ cos θ,r = ρ sinφ

ρ =√x2 + y2 + z2, θ = arccos x

y, φ =

arctan ryv

The Jacobian is ρ2 sinφ.

9 Curvature

Given a vector r, where the derivative is v(velocity), several things can be found.

σ = |v|

T =v

σ

a = σ′T + σ2κN

H = v× a = r′ × r′′

κ = |Hσ3|

N = B × THere, N is the unit normal, T is the unittangent, and B is the unit binormal. If thebinormal is constant, the curve lies in a plane.For the tangential component of acceleration:

at = σ′ = |v · a|

The normal component of acceleration is:

an = a− σ′T

10 Unit Normal

ru × rv|ru × rv|

This can be used to find tangent planes.

11 Grad, Div, and Curl

The gradient of a function is ∇f =<fx, fy, fz >. The Divergence of a function is∇ · f = fx + fy + fz. The curl of a functionis ∇× f. The divergence of curl is always 0.The divergence of the gradiant is the laplacian.

12 Linear Approximation

L(x) = f(a) +∇f(a) · (x− a)

13 Critical Points

A critical point is where ∇(f) = ~0. A point isa maximum if the laplacian (fxxfxy − f2

xy) ispositive and fxx is negative. If the laplacianis positive but fxx is greater than zero, it’s aminimum. If the laplacian is less than zero,it’s a saddle point.

14 Directional Derivatives

Dv = ∇f ·v

|v|

Note: The direction of maximum increaseis the direction of the gradient. Minimumincrease is the opposite of the gradient.

15 Lagrange Multipliers

EXAMPLE: Find the minimum of x2+y2+z2

on the surface 2x+ y+ 2z = 9.

g(x, y, z) = 2x+ y+ 2z − 9 = 0

f(x, y, z) = x2 + y2 + z2

∇f = λ∇g

2x = 2λ

2y = λ

2z = 2λ

2x+x

2+ 2x = 9

Solve for each variable and insert into f forminimum.

16 Double Integrals

Double integrals may be reversed by Fubini’stheorem. Additionally, you may use a changeof variable. This requires multiplication by theabsolute value of the Jacobian. The Jacobianis the ratio of partial derivatives between thevariables used in the change.

∂(x, y)

∂(u, v)=∂x

∂u

∂y

∂v− ∂y

∂u

∂x

∂v

17 Triple Integrals

Basically super-double integrals.

18 Surface Area

∫ ∫R

|~ru ×~rv| dA

∫ ∫R

√f2x + f2

y + 1 dx dy

∫ ∫R

f|∇g|dx dygz

19 Vector Fields

A vector field is said to be conservative if F =∇f for some function f . A vector field cannotbe conservative if curl F 6= ~0. A path mustadditionally be piecewise smooth. If the partialderivatives are continuous, it can be shown that∫

C

F · dr = F(B)− F(A)

20 Green’s Theorem

∮C

P dx+Q dy =

∫ ∫D

(Qx − Py) dA

This can only be used on positively orientedfields. ALWAYS parametrize.

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21 Uses of Green’s Theorem

Many results can be deduced using thetheorem.

1. If Pdx+Qdy is closed, meaningQx−Py =0, then

∫CPdx+Qdy = 0. Thus, it proves

the result about the conservative vectorfields without finding the potential function.

2. For example the area inside the ellipser(t) =< a cos(t), b sin(t) > is easily foundby

12

∫C(xdy−ydx) = 1

2

∫Ca cos(t)(b cos(t)−

b sin(t)(−a sin(t))dt = 12

∫ 2π

0abdt = πab.

Note how the third formula made thework easier!

3. Changing the path without changing theintegral. Suppose that we have Py = Qx ina region D around the origin. Then for anysimple closed positively oriented piecewisesmooth curve C going around the originthe integral

∫CPdx+Qdy is the same.

The idea of the proof is to make considersome small circle totally contained insideC and apply the Green’s theorem to theregion between the two curves, using amild generalization of the theorem.

22 Stoke’s Theorem

∫C

~F · dr =∫ ∫

S

curl ~F · dS

Here dS is ∇g|ω| where ω is the fundamentaldifferential.

ω =dx dy

gz=dy dz

gx=dz dx

gy

23 Surface Integrals.

If a surface S is given in parametric formr =< x(u, v), y(u, v), z(u, v) > where(u, v) ∈ D, we define the integral of a functionf(x, y, z) on S by the formula:∫ ∫

S

f(x, y, z)dS =

∫ ∫D

f(r(u, v))|ru×rv|dA.

However, surfaces don’t usually come witha ready parametrization and it is useful tohave a more general formula handy.

Let the surface S be described by an equa-tion g(x, y, z) = 0, so that we have the basicrelation gxdx+ gydy+ gzdz = 0.

Thus, our integral reduces to∫ ∫S

f(x, y, z)dS =

∫ ∫D

f|∇(g)| dxdy|gz|.

It is easy to see that a similar formula holdswhen gy 6= 0 and we take x, z as parametersand when gx 6= 0 we take y, z as parameters.

We define the Fundamental differentialon S to be:

ωg =dxdy

gz=dy dz

gx=dz dx

gy.

Now we have a single formula for all cases:∫ ∫S

f(x, y, z)dS =

∫ ∫D

f|∇(g)| |ωg|.

Orientable Surface. We note that ∇(g)defines a normal to our surface at all smoothpoints (points where ∇(g) is defined and nonzero.) Also, it varies continuously if g is contin-uous. Then such a surface has a well definedunit normal vector field at all smooth points,

namely n = ∇(g)|∇(g)| .

If our surface encloses a bounded solid (likea sphere, then we can even make sense out ofan outward or inward normal.

We now define the surface integral of avector field F on the surface S by defining it

as the integral of the function F ·n = F · ∇(g)|∇(g)| .

Our formula becomes∫ ∫S

F · n|∇(g)| |ωg| =∫ ∫

D

F · ∇(g) |ωg|.

For a parametric surface, the formulabecomes:∫ ∫

D

f(r(u, v)) · ru × rv dA.

Note that the above formula presumesa certain direction of the unit normal asdetermined by ∇(g) or ru × rv. We multiplyby −1 if we wish to change the direction.

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