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suppose that the velocity vector at a point (x,y,z) in a flowing fluid is given at F(x,y,z) = <x+y,x^3y+16,0>. Find all the stagnation points that is the points where the velocity is 0

Question

suppose that the velocity vector at a point (x,y,z) in a flowing fluid is given at F(x,y,z) = <x+y,x^3y+16,0>. Find all the stagnation points that is the points where the velocity is 0

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Solution

To find the stagnation points, we need to set the velocity vector F(x,y,z) to zero and solve for x, y, and z. The velocity vector is given by F(x,y,z) = <x+y, x^3y+16, 0>.

Setting each component of the vector to zero gives us the following system of equations:

  1. x + y = 0
  2. x^3y + 16 = 0
  3. 0 = 0

The third equation doesn't give us any new information, so we can ignore it.

From the first equation, we can solve for y to get y = -x.

Substitute y = -x into the second equation to get:

x^3(-x) + 16 = 0 -x^4 + 16 = 0 x^4 = 16 x = ±2

Substitute x = 2 and x = -2 into the equation y = -x to get y = -2 and y = 2, respectively.

So, the stagnation points are (2,-2,0) and (-2,2,0).

This problem has been solved

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