Question

If ∈= xyx^- yz²y^-xyz^, evaluate∯ ∈.ds if S is a cube with specific limits for each dimension: 0≤x≤1, 0≤y≤1, and 0≤z≤1. Please calculate the surface integra∯ ∈.ds for this cube S with the given limits.

Answer 1

To evaluate the surface integral ∯ ∈.ds for the given cube with specific limits, we calculate the dot product between the vector field and the unit normal vector at each face of the cube.

Step-by-step explanation:

To evaluate the surface integral ∯ ∈.ds, where ∈= xyx^- yz²y^-xyz^, we need to calculate the dot product between the vector field ∈ and the outward unit normal vector ds. Since S is a cube with dimensions 0≤x≤1, 0≤y≤1, and 0≤z≤1, we can calculate the integral by evaluating the dot product at each face of the cube.

The dot product between the vector field and the outward unit normal vector is given by ∫∫∫ ∈·ds = ∫∫∫ ∈·(∂r/∂x x ∂r/∂y x ∂r/∂z) dxdydz, where r is the position vector.

However, since we have a cube with specific limits for each dimension, we can use the surface area of each face and the value of the vector field to evaluate the integral: ∯ ∈.ds = 2xy + 2yz + 2zx.