Vectors, Vector Fields and Geometry

Question 1

Vector space

Answer 1

If V is a nonempty subset of |R^n, then V will be called a vector space if
For all u , v e V we have u + v e V
For all a e |R and v e V, we have av e V

Question 2

Linearly independent

Answer 2

Let V C |R^n be a vector space.
A set S = {e1, … , ek} C V is linearly independent if
for all a1, .. , ak e |R, such that ∑ aiei = 0, then ai=0 for all i e 1, … , k.

Question 3

Span

Answer 3

If S = { e1, … , ek} C V, then the span of S is the vector space

Span (S) = { ∑(i=1, k) aiei | a1, … , ak e |R.

Question 4

Basis

Answer 4

A set S = { e1, .. , ek} C V is a basis of V if S is linearly independent and the span of S is V.

Question 5

The dot product

Answer 5

Let u = (x1, x2, x3) and v = (y1, y2, y3) be vectors in |R^3.
The dot product is defined by
u.v = x1y1 +x2y2 + x3y3

Question 6

The cross product

Answer 6

Let u = (x1, x2, x3) and v = (y1, y2, y3) be vectors in |R^3. The cross product
u x v = |i j k |
|x1 x2 x3|
|y1 y2 y3|
=(x2y3 - x3y2)i + (x3y1 - x1y3)j + (x1y2 - y1x2)k

Question 7

Standard properties of two products

Answer 7

For u, v e |R^3

1) u.v =v.u
2) |u.v| = |u||v|cosa (where a is the angle between the two vectors)
3) u.v =0 => u is perpendicular to v
4) |v|^2 = |v .v|
5) The projection of v onto u is (v . u )/ (|u|^2) u
6) u is perpendicular to (u x v) and v is perpendicular to (u x v)

Question 8

Standard properties (cross product)

Answer 8

7) u x v = -v x u
8) |u x v| = |u| |v| sin a
9) |u x v| equals area of the parallelogram determined by the two vectors
10) u . (v x w) = ( u x v ) . w
11) |u . (v x w)| is the volume of the parallelepiped determined by the three vectors
12) u x v = 0 => u = av for some scalar a e |R

Question 9

Gradient

Answer 9

Suppose u : Ω -> |R is a differentiable scalar field. If Ω C |R^3 then the gradient of u is the vector field
▽u = ∂u/∂x i + ∂u/∂y j + ∂u/∂z k.

Question 10

Laplacian

Answer 10

The Laplacian of u is the scalar fields ▽^2u = △u = ∂^2u/∂x^2 + ∂^2u/∂y^2 + ∂^2u/∂z^2

Question 11

Divergence

Answer 11

Let Ω C |R^3 and let u : Ω -> |R^3,
u(x, y, z) = u1(x , y, z)i + u2 (x, y, z) j + u3 (x, y, z)k be a vector field.
The divergence of u is the scalar field
div u = ▽ . u = ∂u1/∂x + ∂u2/∂y + ∂u3/∂z

Geometrically - measures how much the vector field streams out from a given point

Question 12

Curl

Answer 12

The curl of u is the vector field
curl u = ▽ x u = | i j k |
|∂/∂x ∂/∂y ∂/∂z|
| u1 u2 u3 |
= (∂u3/∂y - ∂u2/∂z) i + (∂u1/∂z - ∂u3/∂x) j + (∂u2/∂x - ∂u1/∂y)k

Geometrically - measures how much the field swirls around a certain point

Question 13

Divergence and curl properties

Answer 13

1) ▽ . ( u + v ) = ▽ . (u) +▽. (v) , that is div(u+v) = div(u) + div(v).
2) ▽ x (u + v) = ▽ x(u) + ▽ x (v), that is curl (u+v) = curl(u) + curl(v)
3) ▽ . (wu) = ▽w . u = w ▽.u (where w is a scalar)
4) ▽ x (wu) = ▽w x u = w ▽x u (where w is a scalar)
5) curl (▽w) = ▽ x ▽w =0
6) div curl u = ▽ .( ▽ x u ) = 0
7) div ▽w = ▽^2w