Finding the Curl of a Vector Field at a Specific Point

In vector calculus, the curl of a vector field is a vector field that describes the infinitesimal rotation of the vector field around a given point. The curl of a vector field F = (f, g, h) is defined as:

curl F = (∂h/∂y - ∂g/∂z, ∂f/∂z - ∂h/∂x, ∂g/∂x - ∂f/∂y)

where ∂/∂x, ∂/∂y, and ∂/∂z represent the partial derivatives with respect to x, y, and z, respectively.

Let's apply this concept to find the curl of the vector field F = z sin(x) i - 2x cos(y) j + y tan(z) k at the point (π, 0, π/4).

1. Step 1: Identify the vector field components.

• f(x, y, z) = z sin(x)
• g(x, y, z) = -2x cos(y)
• h(x, y, z) = y tan(z)

1. Step 2: Calculate the partial derivatives.

• ∂h/∂y = tan(z)
• ∂g/∂z = 0
• ∂f/∂z = sin(x)
• ∂h/∂x = 0
• ∂f/∂y = 0
• ∂g/∂x = -2 cos(y)

1. Step 3: Substitute the values into the curl formula.

• curl F = (∂h/∂y - ∂g/∂z, ∂f/∂z - ∂h/∂x, ∂g/∂x - ∂f/∂y)
• curl F = (tan(π/4) - 0, sin(π) - 0, -2 cos(0) - 0)
• curl F = (tan(π/4), -1, -2)

1. Step 4: Evaluate the curl at the given point.

• At the point (π, 0, π/4), the curl of the vector field F is (tan(π/4), -1, -2).

In conclusion, the curl of the vector field F = z sin(x) i - 2x cos(y) j + y tan(z) k at the point (π, 0, π/4) is (tan(π/4), -1, -2).