Lectures 7, 8 and 9 Flashcards
What is an important equation for velocity in cylindrical polars?
v = dρ/dt * eρ + ρ*dϕ/dt * eϕ + dz/dt * ez
What is x, y and z equal to in spherical polar coordinates?
x = rsinθcosϕ y = rsinθsinϕ z = rcosθ
In spherical polars, what are θ and ϕ measured between? Which plane is ϕ measured in? What is θ a measurement of?
0 <= θ <= π
0 <= ϕ <= 2π
ϕ is measured in the x-y plane.
θ is the angle from the z-axis we are looking at
Define the changes in surface and volume, ds and dv for spherical polars.
ds = r^2 * sinθ dϕdθ dv = r^2 * sinθ drdϕdθ
What are the three basis vectors for spherical polars?
er, eϕ and eθ
What is the equation for the basis vector er?
er = cosϕsinθ i + sinϕsinθ j + cosθ k
What is the equation for the basis vector eϕ?
eϕ = - i + cosϕ j
What is the equation for the basis vector eθ?
eθ = cosθcosϕ i + cosθsinϕ j - sinθ k
For a 1D integral, what is the change in step ds equal to?
ds^2 = dx^2 + dy^2
What is an alternative version to write ds?
ds = dx*sqrt(1+(dy/dx)^2), or with dy out the front and dx/dy in the brackets
What is a Jacobian?
The transformation from denoting coordinates as one thing to another.
Give an example of a Jacobian.
The Jacobian of x,y,z with respect to u,v,w is: J = d(x,y,z)/d(u,v,w)
What can we write out change in volume element dv as including the Jacobian?
The magnitude of the Jacobian multiplied by the changed letters: dv = dxdydz = |J|dudvdw
How would you calculate the Jacobian?
It is the determinant including the variables involved. e.g. if J = d(x,y)/d(ρ,ϕ), then it is the determinant of these components.
Why is the Jacobian useful?
It is an easier way of working out the coordinate system transformations for dA and dV etc, by using the determinant and substituting in the corresponding conversion equations.
What are hyperbolic coordinates?
Where you convert the coordinates by using the two axis, for example x,y, and using xy = r, and x/y = s, giving a simple square.
In a surface of revolution, what is the change in area dA equal to?
dA = dsydθ, where ds is an element along the line revolved, y is the height of the line, and dθ is the angle of revolution.
How do you find the total surface area of revolution if it is revolved by 2π radians?
A = integral from s1 to s2 of 2πyds, where s1 and s2 are the start and end points of the line/curve
What is the equation for the centroid of y, the function we are revolving?
y(centroid) = integral of y*ds / integral of ds
How do you get an equation for area using the centroid equation?
Multiply each side by 2π, and then you have A/S, then rearrange for A.
What is Pappus’ 2nd Theorem?
A = 2π * y(centroid) * s
