Mathematics for Physicists Test Revision Flashcards
What is the formula for the dot product?
u⋅v = |u||v|cosθ
What is the cross product formula?
uxv = |u||v|sinθ n
where in is the unit vector perpendicular to u and v
How can you tell if two vectors, u and v, are perpendicular?
u⋅v = 0
What is the integral of tanx?
ln(cosx)
What is the differential of tanx?
sec^2(x)
What is the format for an equation required for the integrating factor method to be used?
dy/dx + P(x)y = Q(x)
What is the formula for finding R(x) in the integrating factor method?
R(x) = e^(∫P(x)dx)
What is the formula that can be used to solve the equation once R(x) has been found in the integrating factor method?
d/dx(R(x)y) = R(x)Q(x)
What is the Taylor series formula?
(f^(n’) (a))/n!) *(x - a)^n
What is the formula to find each reciprocal vector for a set of three orthogonal vectors, a, b and c?
a’ = (bxc)/a⋅(bxc)
b’ = (cxa)/a⋅(bxc)
c’ = (axb)/a⋅(bxc)
What can reciprocal vectors be used for?
They can be dotted with a given vector to find the components of that vector with respect to original basis vectors
What is the formula for the volume of a parallelepiped with sides with vectors a, b, c?
a⋅b⋅(a⋅c x a⋅d)
What is the integral of arctanx?
1/(1+x)
What is the differential of arctanx
1/(1+x^2)
What does an under damped curve look like?
An sine wave with exponentially decreasing amplitude
What does a over damped curve look like?
A nearly straight line with a shallow negative gradient
What does a critical damping curve look like?
A line that decreases quickly to perfectly meet the x axis, with a sinusoidal shape to it
How to identify a saddle point?
fxxfyy - fxy^2 < 0
How to identify a local maximum?
fxxfyy - fxy^2 > 0 and fxx < 0
How to identify a local minimum?
fxxfyy - fxy^2 > 0 and fxx > 0
What happens if fxxfyy - fxy^2 = 0?
The test is inconclusive
