Determine whether the following vector field is incompressible or irrotational at the point (0,1,2).
Vector F = [x/x^2+y^2+z^2]i^+[y/x^2+y^2+z^2]j^+[z/x^2+y^2+z^2]k^
incompressible = solenoiodal iff ∇•F = 0
∇•F = ∂F/∂x + ∂f/∂y + ∂f/∂z
= (-x^2+y^2+z^2)/(x^2+y^2+z^2)^2
+ (x^2-y^2+z^2)/(x^2+y^2+z^2)^2
+ (x^2+y^2-z^2)/(x^2+y^2+z^2)^2
= 1/(x^2+y^2+z^2)
∇•F at (0,1,2) = 1/5 ≠ 0
irrotational iff ∇×F = 0
I leave it to you to verify that in fact ∇×F = 0.
To determine whether a vector field is incompressible or irrotational at a given point, we need to calculate the divergence and curl of the vector field at that point.
Let's start by calculating the divergence of the vector field F = [x/(x^2+y^2+z^2)]i + [y/(x^2+y^2+z^2)]j + [z/(x^2+y^2+z^2)]k. The divergence is given by the formula:
div(F) = ∂F/∂x + ∂F/∂y + ∂F/∂z
Taking partial derivatives of each component, we have:
∂F/∂x = ∂(x/(x^2+y^2+z^2))/∂x = 1/(x^2+y^2+z^2) - 2x^2/(x^2+y^2+z^2)^2
∂F/∂y = ∂(y/(x^2+y^2+z^2))/∂y = 1/(x^2+y^2+z^2) - 2y^2/(x^2+y^2+z^2)^2
∂F/∂z = ∂(z/(x^2+y^2+z^2))/∂z = 1/(x^2+y^2+z^2) - 2z^2/(x^2+y^2+z^2)^2
Now, substituting x = 0, y = 1, and z = 2, we can calculate the divergence at the point (0, 1, 2):
div(F) = 1/(0^2+1^2+2^2) - 2(0^2)/(0^2+1^2+2^2)^2 + 1/(0^2+1^2+2^2) - 2(1^2)/(0^2+1^2+2^2)^2 + 1/(0^2+1^2+2^2) - 2(2^2)/(0^2+1^2+2^2)^2
Simplifying the expression, we get:
div(F) = 1/5 - 0/5^2 + 1/5 - 2/5^2 + 1/5 - 8/5^2
div(F) = 1/5 + 1/5 - 8/5^2
div(F) = 2/5 - 8/25
div(F) = 10/25 - 8/25
div(F) = 2/25
Therefore, at the point (0, 1, 2), the divergence of the vector field F is 2/25.
To determine whether the vector field F is incompressible or irrotational at the point (0, 1, 2), we need to analyze the result:
- If the divergence is zero (div(F) = 0), the vector field is incompressible at that point.
- If the curl is zero (curl(F) = 0), the vector field is irrotational at that point.
Since the divergence of the vector field F is 2/25, which is not zero, we can conclude that the vector field F is not incompressible at the point (0, 1, 2).
To determine whether it is irrotational, we need to calculate the curl of the vector field F, but the curl is not needed in this case since the vector field is already determined to be not incompressible.