MAS211 Flashcards
Fourier Series
a0/2 + sum(1 to inf) [an cos(nx) + bn sin(nx)]
Fourier Coefficients
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
What are odd/even functions? What are the sine/cosine Fourier series?
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)
Define div
dv1/dx + dv2/dy + dv3/dz
partial derivatives
Define curl
(dv3/dy - dv2/dz) i +
(dv1/dz - dv3/dx) j +
(dv2/dx - dv1/dy) k
Stokes’ Theorem
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
Normal vector to a surface
N = dr/du x dr/dv
n = N/|N|
Divergence Theorem (Gauss’ Theorem)
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
Green’s Theorem
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
Dirichlet Conditions
- 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.
- f(x) must have only a finite number of maxima and minima on the interval.
Chain Rule for GoF
D(G o F)(u, v) = D(G)(F(u, v))D(F)(u, v)
Kernel
ker(L) = {u in real | L(u) = 0q}
Image
im(L) = {x in real | x = L(u) for some u in real}
Nullity
The nullity of L is the dimension of ker(L).
Rank
The rank of L is the dimension of im(L).
The Rank-Nullity Theorem
null(L) + rank(L) = p.
Conditions for vector subspace
0 in V
closed
multiply by t
Polar coordinates
x = r cos theta y = r sin theta
Linear map
L(x + y) = L(x) + L(y)
L(tx) = tL(x).
Basis
A list is a basis for V if it spans V and is linearly independent.
Span
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)
Dimension
The dimension of V is the number of elements in a basis of V .
sin(2x) and cos(2x) identities
sin(2x) = 2sin(x)cos(x) cos(2x) = 1 - 2sin^2(x)
Classifying Stationary Points
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.
Linear Independence
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.
Derivative of tan inverse
1 / (1+x^2)
Derivative of sin inverse
1 / sqrt(1 -x^2)
Derivative of cos inverse
-1 / sqrt(1 -x^2)
