Lectures 10, 11 and 12 Flashcards
What is the change in volume element dV when an area element is revolved by dθ?
dV = dAydθ
What is the integral equation for the volume of revolution?
V = π * integral from x1 to x2 of f(x)^2 dx
What is Pappus’ first theorem?
V = 2πAy(centroid)
What is the divergence of a vector and what is the divergence symbol?
∇.v, where v is a vector. ∇ = d/dx i + d/dy j + d/dz k
How do we interpret the divergence of say an electric field when it is equal to zero?
Net flux of field lines through a volume - same number of lines enter as leave so is zero.
What is the divergence of a magnetic field in all space?
Zero.
What is the curl of a vector and how is it calculated?
∇ X v, where v is a vector. Equals the determinant with the three vector components along the bottom, i,j,k along the top and d/dx, d/dy and d/dz along the middle.
What are two cases with a raft in water where the curl is and is not equal to zero respectively? What happens when the curl is not equal to zero?
Equals zero when v = x i
Not equal to zero when v = x j - in this case the raft will tend to turn to the left.
What is the curl equal to for a whirlpool? What happens to a paddle wheel if you put it in the whirlpool?
Zero. A paddle wheel will circulate around the centre but will not spin.
What is grad squared also known as?
The Laplacian.
What is Laplace’s equation?
∇^2 * Ф = 0
What is Poisson’s equation?
∇^2 * Ф = k
What is curl grad phi? How is it calculated?
∇ X ∇*Ф. Calculated with the determinant but along the bottom is the differentiated vector components rather than normal.
What is curl grad phi always equal to?
Zero.
What is div curl A and how is it calculated?
∇*(∇ X A). Calculated by doing the curl part first then taking divergence of each component.
What is div curl A equal to?
Zero.
What is the gradient of a divergence and how is it calculated?
∇*(∇ * A). Calculated by taking the divergence of a first and then multiplying it all out.
What is the gradient of a divergence equal to?
A vector.
What is curl curl A?
∇ X (∇ X A)
What is another way of writing curl curl A?
∇^2 A = ∇*(∇ * A) - ∇ X (∇ X A)
How can we calculated ∇*(fg), where f and g are scalars?
Use the product rule:∇(fg) = f(∇g) + g(∇*f)
How can we calculate ∇*(fg), where f is a scalar and g is a vector?
∇(fg) = g * ∇f + f(∇*g)
