Review of Multivariable Calculus, Integral Transforms, and Parameter Dependent Integrals

Vocabulary flashcards from a lecture transcript on functions of several variables, parametric integrals, and integral transformations.

N1 norm

Norm that is the sum of the absolute values.

N2 norm

Norm that is the square root of the sum of the squares.

N infinity norm

Norm that is the max of the absolute values.

Definition of Continuity

Limit of f(x) as x approaches a is f(a).

Polar Coordinates

x = r cos(theta), y = r sin(theta); used to verify limit at 0,0.

Uniform Convergence Theorem

If a sequence of continuous functions converges uniformly to f, then f is continuous.

Partial Derivatives

Derivatives calculated by considering other variables as constants.

Class Ck

Partial derivatives up to order k exist and are continuous.

Schwartz's Theorem

For class Ck with k >= 2, the order of differentiation doesn't matter.

Differentiability

f(a + h) = f(a) + L(h) + o(h), where L(h) is the differential DfA.

Sufficient Condition for Differentiability

If F is of class C1 on an open set, then F is differentiable everywhere on that open set.

Differential DfA (Real-Valued Function)

For a function from Rn to R, it's the dot product of the gradient with the vector h.

Differential DfA (Vector-Valued Function)

Represented by the Jacobian matrix.

Chain Rule for Differentials

The differential of the composite is the composite of the differentials.

Critical Point

Gradient of f(a) equals zero.

Hessian Matrix Sign

Defined positive: local min; negative: local max; indefinite: saddle point.

Lagrange Multipliers

Form the Lagrangian L(x, lambda) = f(x) + lambda g(x) and find points where the gradient of L w.r.t. x and lambda is zero.

Laplacian

Delta f = sum of (partial^2 f / partial xi^2).

Simple Convergence

Study of improper integral for a fixed x.

Uniform Convergence

Convergence with respect to x.

Weierstrass M-Test

Find an integrable function phi(t) independent of x that bounds f(t, x) for all x in an interval.

Continuity of Integral (Segment)

If f(t, x) is continuous, then f is continuous.

Sufficient Condition for Continuity (Improper Integral)

Domination by integrable phi(t).

Leibniz's Theorem

f'(x) = integral of (partial f / partial x)(t, x) dt.

Gaussian Integral G(0)

integral from 0 to +infinity of e^(-t^2) dt = sqrt(pi)/2.

Fourier Transform

Tool for changing from time to frequency.

Laplace Transform

Tool for changing to the complex plane.

Fourier Transform Definition

F(f) = (1/sqrt(2pi)) * integral from -infinity to +infinity of f(t) * exp(-ixt) dt.

Fourier Transform of Derivative

F(f') = ixF(f).

Convolution Theorem (Fourier)

Convolution in the time domain becomes multiplication in the frequency domain.

Inverse Fourier Transform

f(t) = (1/sqrt(2pi)) * integral from -infinity to +infinity of F(x) * exp(itx) dx.

Parseval's Equality

integral of |f(t)|^2 dt = integral of |F(x)|^2 dx.

Laplace Transform Definition

L{f(p)} = integral from 0 to +infinity of exp(-pt) * f(t) dt.

Laplace Transform of Derivatives

L{y'(p)} = pY(p) - y(0); L{y''(p)} = p^2Y(p) - py(0) - y'(0).

Convolution Theorem (Laplace)

L{f * g} = L{f(p)} * L{g(p)}.