MAS211

Question 1

Fourier Series

Answer 1

a0/2 + sum(1 to inf) [an cos(nx) + bn sin(nx)]

Question 2

Fourier Coefficients

Answer 2

a0 = 1/pi integral(pi to -pi) f(x) dx
an = 1/pi integral(pi to -pi) f(x) cos(nx) dx
bn = 1/pi integral(pi to -pi) f(x) sin(nx) dx

Question 3

What are odd/even functions? What are the sine/cosine Fourier series?

Answer 3

f(-x) = f(x) - even
f(-x) = -f(x) - odd
If f(x) is even: f(x) = a0/2 + sum(1 to inf) an cos(nx)
If f(x) is odd: f(x) = sum(1 to inf) bn sin(nx)

Question 4

Define div

Answer 4

dv1/dx + dv2/dy + dv3/dz

partial derivatives

Question 5

Define curl

Answer 5

(dv3/dy - dv2/dz) i +
(dv1/dz - dv3/dx) j +
(dv2/dx - dv1/dy) k

Question 6

Stokes’ Theorem

Answer 6

Let S be a piecewise smooth oriented surface that is bounded by a simple, closed piecewise smooth curve C with positive orientation. If v(x; y; z) is a continuous vector field with continuous 1st partial derivatives on some open set containing S, then
∫(C) v.dr = ∫(S) curl(v).n

Question 7

Normal vector to a surface

Answer 7

N = dr/du x dr/dv

n = N/|N|

Question 8

Divergence Theorem (Gauss’ Theorem)

Answer 8

Let R be a region of R3 bounded by a closed simple surface S, which is oriented outwards. If v(x; y; z) is a vector field whose components have continuous
partial derivatives on some open set containing R, and if ^n(x; y; z) is the outward-directed normal at position (x; y; z) on S, then
∫ v.n dS = ∫ div(v) dV

Question 9

Green’s Theorem

Answer 9

Let R be a simply connected closed region of the plane whose boundary is a simple, piecewise smooth, closed curve C oriented anticlockwise. If f(x; y) and g(x; y) are continuous and have continuous 1st partial derivatives on some open set containing R, then
∫(C) f(x,y) dx + g(x,y) dy = ∫∫(R) (dg/dy - df/dx) dx dy

Question 10

Dirichlet Conditions

Answer 10

1. f(x) must be single-valued and continuous on the interval, except for at a finite number of points at which it has finite discontinuities.
2. f(x) must have only a finite number of maxima and minima on the interval.

Question 11

Chain Rule for GoF

Answer 11

D(G o F)(u, v) = D(G)(F(u, v))D(F)(u, v)

Question 12

Kernel

Answer 12

ker(L) = {u in real | L(u) = 0q}

Question 13

Image

Answer 13

im(L) = {x in real | x = L(u) for some u in real}

Question 14

Nullity

Answer 14

The nullity of L is the dimension of ker(L).

Question 15

Rank

Answer 15

The rank of L is the dimension of im(L).

Question 16

The Rank-Nullity Theorem

Answer 16

null(L) + rank(L) = p.

Question 17

Conditions for vector subspace

Answer 17

0 in V
closed
multiply by t

Question 18

Polar coordinates

Answer 18

x = r cos theta
y = r sin theta

Question 19

Linear map

Answer 19

L(x + y) = L(x) + L(y)

L(tx) = tL(x).

Question 20

Basis

Answer 20

A list is a basis for V if it spans V and is linearly independent.

Question 21

Span

Answer 21

A set of elements of V spans V if every element in V can be written as a linear combination of the set of elements.
(set)

Question 22

Dimension

Answer 22

The dimension of V is the number of elements in a basis of V .

Question 23

sin(2x) and cos(2x) identities

Answer 23

sin(2x) = 2sin(x)cos(x)
cos(2x) = 1 - 2sin^2(x)

Question 24

Classifying Stationary Points

Answer 24

If A > 0 and AC > B^2, then local minimum.
If A < 0 and AC > B^2, then P is a local maximum.
If AC < B^2, then P is a saddle point.
If either AC = B2, or if A = 0;B = 0 and C = 0, then no conclusion.

Question 25

Linear Independence

Answer 25

A set {x1, x2, …, xn} is linearly independent iff the only scalars {t1, t2, …, tn} such that x1t1 + … + xntn = 0 are t1 = t2 = … = tn = 0.

Question 26

Derivative of tan inverse

Answer 26

1 / (1+x^2)

Question 27

Derivative of sin inverse

Answer 27

1 / sqrt(1 -x^2)

Question 28

Derivative of cos inverse

Answer 28

-1 / sqrt(1 -x^2)