Integration Flashcards
What are the expressions for x and y in polar coordinates?
x = rcos(θ) y = rsin(θ)
What is the area element dA equal to in polar coordinates?
dA = r dr dθ
How many dimensions are polar coordinates?
2 dimensions
What are cylindrical coordinates?
Polar coordinates extended into 3 dimensions, adding the cartesian z coordinate.
What are the expressions for x, y and z in cylindrical coordinates?
x = rcos(θ) y = rsin(θ) z = z
What is the volume element dV in cylindrical coordinates?
dV = r dθ dr dz
What are the 3 components in spherical coordinates and what do they represent?
- r = distance from the origin
- θ = angle away from z axis subtended by OP (clockwise between 0 and pi)
- ϕ = angle from x axis subtended by projection of vector OP onto y axis (anticlockwise between 0 and 2pi)
What are the expressions for x, y and z in spherical coordinates?
x = rsinθcosϕ y = rsinθsinϕ z = rcosθ
What is the volume element dV in spherical coordinates?
dV = r^2 sinθ dr dθ dϕ
How do you construct a line integral?
Parameterise the curve r to get |dr/dλ| dλ, where |dr/dλ| is the derivatives of x, y, z etc added together and square rooted
How do you construct a line integral for a vector field?
int (F.dR) = int (Fxi + Fyj + Fzk) . (dxi + dyj + dzk)
= int (Fx dx + Fy dy + Fz dz)
If you have an equation (e.g. x = y^2) then find the derivative of the equation and then use this in the above formula. Then pick either dy or dx and integrate between the change in y or the change in x.
What is the surface change dS for spherical coordinates and cylindrical coordinates?
Spherical - dS = r^2 sinθ dθ dϕ
Cylindrical - dS = r dr dθ
What are surface integrals used for?
Calculate the surface area.
How do you perform surface integrals involving vectors?
dS (vector) = dS r(hat-vector)
Basically add an r(hat) into the integral
How do you calculate the gradient of a function ϕ?
dϕ/dx i + dϕ/dy j + dϕ/dz k
where d is a partial derivative.
