Multivariable Calculus Flashcards
Define Euclidean distance for x,y in Rn
Define the Euclidean norm
Define the | . |1 norm
Define convergence for a sequence of vectors (xj)
Define the scalar product
State and prove the Cauchy-Schwartz inequality
Define cos theta with regards to the cauchy schwartz inequality
State and prove the triangle inequality
State the relationship between the euclidean norm and the 1 norm
|x| <= |x|1 <= sqrt(n) |x|
Define the infinity norm
State and prove the relationship between the euclidean norm and the infinity norm
Prove the uniqueness of limits for a sequence (xj)
Give the sequential definition of continuity
f is continuous at p, if for every sequence (xj) which converges to p, f(xj) converges to f(p)
Prove that a Cauchy sequence (xj) is convergent
Define the Open Ball
Define continuity of a function f: U to Rn at p in terms of open balls
When is U, a subset of Rn, open?
Define the epsilon nieghbourhood of E
Proposition: If E1,……., Em are all closed then the union is closed
Proposition: Let U1,……,Um be open sets, then the intersection is open
Proposition: A set is closed if and only if it contains all its limit points
Define relatively open
Define an isolated point of U
Define a continuous limit
Define a path from p in Rn to q in Rn
A path is a continuous map r: [a,b] to U, [a,b] in R such that r(a) = p and r(b) = q
Define path connected for U, a subset of Rn
for all p,q in U, there is a path r:[a,b] to U such that r(a) = p and r(b) = q
Define sequential compactness for K, a subset of Rn
K, a subset of R2 is sequentially compact if and only if K is closed and bounded
State and prove the extreme value theorem a subset K in Rn
Let V = { (x,y) : xy not equal to 0}. Prove V isn’t path connected
What is meant by L(Rn, Rk) and M(k x n, R)
Define || (aij) ||2
Define the operator norm
What is the relationship between || A || and || (aij) ||2?
Prove that ||BA|| <= ||B|| ||A||
Define the General Linear group
Define differentiability for a point p, in U, a subset of Rn
State the chain rule
Define directional derivative
Define the partial derivative
Define the Jacobian matrix
What condition must hold in the Jacobian matrix for p to be differentiable
it must exist at all points
Define continuously differentiable
What is the Jacobian form of the chain rule?
Define grad f
Given f: Rn to R and g:R to R whats the jth partial derivative of g(f(x)), and then define grad g(f(x))
define d/dt f(r(t)) for f:Rn to R and r:R to Rn
Define a change of variables
State the Inverse function theorem
State the Implicit function theorem
Define a vector field
Define the Curve Cpq
Define the tangential line integral
Define a gradient field
If a vector field v is the gradient of a function f : U → R then v is called a gradient field.
State the fundamental theorem of calculus for a gradient vector field
0
Define conservative
Theorem: A vector field v: U to Rn is a gradient field if and only if its conservative
When is f called a scalar potential of v
When v = grad f
Define v perp, and then the normal to a curve C
Define the flux of a vector field in R2
Derive greens theorem for a rectangle
Define a region in Rn
Define curl
Define a positively oriented regular parameterisation
State Greens theorem for a planar region
Give the equation for the flux of of v across the boundary of omega
Define the divergence of a vector field
State the divergence theorem
Give the two equations for the flux of v across a surface S in R3
State the divergence theorem in R3
Define a radial function
Define Hess f(p)
When do second order partial derivatives commute?
D2f exists
the second partial derivatives are continuous
State Taylor’s theorem, both for the 1 variable case, and otherwise
State the conditions for when a symmetric matrix is
i) positive definite
ii) positive semidefinite
iii) negative definite
iv) negative semidefinite
v) indefinite
State the second order derivative test
State the definitiveness test for 2x2 symmetric matrices
