theorems to memorise Flashcards
what is the dot product
the scalar product
u.v = mod(uv)cos(theta)
u1v1 + u2v2 + u3v3
if the dot product is 0 they are orthogonal(right angles to each other)
what is the vector cross product
make a matrix where its [I,j,k][u1,u2,u3][v1,v2,v3]
the cross product is the determinant
uw = -(wu)
u*u = 0
what is the scalar triple product
u.(v*w)
how do differentiability and continuity relate
If a function f is differentiable at x = a then f is continuous at x = a.
what is the second definition of differentiation from first principle
lim x->a
(f(x)-f(a))/(x-a)
what does tanh(x) differentiate to
tanhx = 1−tanh^2(x)
what is the formula for the second derivative of a parametric equation
dy/dx differentiated again, divided by dx/dt
what is Leibniz rule
sum(nCr)f(r’th derivative)g((n-r)th derivative) = dny/dxy
What is Rolles theorem
Suppose f(x) is continuous for all x ∈ [a, b], differentiable for all x ∈ (a,b), and f(a) = f(b). Then there exists a value c ∈ (a, b) such that
f′(c) = 0.
what is the mean value theorem
If f (x) is continuous for all x ∈ [a, b] and differentiable for all x ∈ (a,b) then there exists a c ∈ (a,b) such that
f′(c)= f(b)−f(a)/b−a
what is Leibniz test
if a series is in the form (-1)^n(An)
if an is greater than 0, decreasing and tends to 0 as n tends to infinity, it converges
what’s a power series
(An)x^n
the radius of convergence is
1/P when between 0 and infinity, 0 at infinity and infinite at 0
what is polar form multiplication
z1z2 = r1r2 (cos(θ1 +θ2)+isin(θ1 +θ2)).
what is de moires theorem
z^r =
r^n(cos(ntheta) + Isin(ntheta))
ln(z) = ln |z| + i arg(z)
ln(z) = ln |z| + i arg(z)
