To determine if a vector field is solenoidal, we need to check if the divergence of the vector field is zero. The divergence of a vector field F = (P, Q, R) is given by ∇.F = ∂P/∂x + ∂Q/∂y + ∂R/∂z.

1. For E = (x^3*y^2, -y^3*x^2, x^3*y^3), the divergence is:

∇.E = ∂(x^3*y^2)/∂x + ∂(-y^3*x^2)/∂y + ∂(x^3*y^3)/∂z
= 3x^2*y^2 - 3y^2*x^2 + 0
= 0

So, E is solenoidal.

2. For B = (e^x, e^y, e^z), the divergence is:

∇.B = ∂(e^x)/∂x + ∂(e^y)/∂y + ∂(e^z)/∂z
= e^x + e^y + e^z

Since this is not equal to zero, B is not solenoidal.

So, the vector field E is solenoidal and B is not.

Question

To determine if a vector field is solenoidal, we need to check if the divergence of the vector field is zero. The divergence of a vector field F = (P, Q, R) is given by ∇.F = ∂P/∂x + ∂Q/∂y + ∂R/∂z.

1. For E = (x^3*y^2, -y^3*x^2, x^3*y^3), the divergence is:

∇.E = ∂(x^3*y^2)/∂x + ∂(-y^3*x^2)/∂y + ∂(x^3*y^3)/∂z
     = 3x^2*y^2 - 3y^2*x^2 + 0
     = 0

So, E is solenoidal.

2. For B = (e^x, e^y, e^z), the divergence is:

∇.B = ∂(e^x)/∂x + ∂(e^y)/∂y + ∂(e^z)/∂z
     = e^x + e^y + e^z

Since this is not equal to zero, B is not solenoidal.

So, the vector field E is solenoidal and B is not.

Knowee AI · Accepted Answer

To determine if a vector field is solenoidal, we need to check if the divergence of the vector field is zero. The divergence of a vector field F = (P, Q, R) is given by ∇.F = ∂P/∂x + ∂Q/∂y + ∂R/∂z.

1. For E = (x^3*y^2, -y^3*x^2, x^3*y^3), the divergence is:

∇.E = ∂(x^3*y^2)/∂x + ∂(-y^3*x^2)/∂y + ∂(x^3*y^3)/∂z
     = 3x^2*y^2 - 3y^2*x^2 + 0
     = 0

So, E is solenoidal.

2. For B = (e^x, e^y, e^z), the divergence is:

∇.B = ∂(e^x)/∂x + ∂(e^y)/∂y + ∂(e^z)/∂z
     = e^x + e^y + e^z

Since this is not equal to zero, B is not solenoidal.

So, the vector field E is solenoidal and B is not.

Suppose E = (x3y2, −y3x2, x3y3) and B = (ex, ey, ez ). Determine which (if any) vector ﬁeldis solenoidal and explain why.[5 marks]END OF PAPER3

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