Question

Determine ∇×A of each of the following vector fields:

A= x(4xsin4y) - (4cos4y), for − π ≤ x , y ≤ π

Answer 1

The curl of the vector field A, which is defined in the domain −π ≤ x, y ≤ π, is computed to be -16x^2cos(4y)k.

Step-by-step explanation:

To determine the curl of the vector field A given as A = x(4xsin4y) - (4cos4y), we need to use the curl operator in a Cartesian coordinate system which is defined as:

∇ × A = ∇ × (Ax, Ay, Az) = ∇ × (4x2sin(4y), -4cos(4y), 0)

The curl of a vector field in three dimensions is given by:

∇ × A = ∇ × (Ax, Ay, Az) =

i ∂Az/∂y - ∂Ay/∂z) -
j (∂Az/∂x - ∂Ax/∂z) +
k (∂Ay/∂x - ∂Ax/∂y)

Since the vector field has no z-component, this simplifies to:

∇ × A = k (∂Ay/∂x - ∂Ax/∂y)

Performing the partial derivatives, we find:

∂Ay/∂x = -4 × 0 = 0

∂Ax/∂y = ∂/(4x2sin(4y))/∂y = 16x2cos(4y)

Thus, the curl of A is:

∇ × A = 0 - 16x2cos(4y) = -16x2cos(4y)k

Note that since the vector field is defined for −π ≤ x, y ≤ π, the curl is also valid in this domain.