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Vector Fields

Vector Fields. 1. Computer ||F|| and sketch several representative vectors in the vector field (Similar to p.1067 #7-16). 2. Find the conservative vector field for the potential function by finding its gradient (Similar to p.1067 #21-29) Hint: F(x, y) = f x i + f y j.

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Vector Fields

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  1. Vector Fields

  2. 1. Computer ||F|| and sketch several representative vectors in the vector field(Similar to p.1067 #7-16)

  3. 2. Find the conservative vector field for the potential function by finding its gradient(Similar to p.1067 #21-29)Hint: F(x, y) = fxi + fyj

  4. 3. Find the conservative vector field for the potential function by finding its gradient(Similar to p.1067 #21-29)Hint: F(x, y) = fxi + fyj

  5. 4. Verify that the vector field is conservative(Similar to p.1067 #31-34)Hint: F(x, y) = Mi + Njconservative if:

  6. 5. Determine whether the vector field is conservative.(Similar to p.1067 #35-38)Hint: F(x, y) = Mi + Njconservative if:

  7. 6. Determine whether the vector field is conservative.(Similar to p.1067 #35-38)Hint: F(x, y) = Mi + Njconservative if:

  8. 7. Determine whether the vector field is conservative. If it is, find a potential function for the vector field(Similar to p.1067 #39-48)Hint: F(x, y) = fxi + fyjafter you get fx and fy, take the integral with respect to the variable to get the potential function

  9. 8. Determine whether the vector field is conservative. If it is, find a potential function for the vector field(Similar to p.1067 #39-48)Hint: F(x, y) = fxi + fyjafter you get fx and fy, take the integral with respect to the variable to get the potential function

  10. 9. Determine whether the vector field is conservative. If it is, find a potential function for the vector field(Similar to p.1067 #39-48)Hint: F(x, y) = fxi + fyjafter you get fx and fy, take the integral with respect to the variable to get the potential function

  11. 10. Find curl F for the vector field at the given point(Similar to p.1067 #49-52)

  12. Test for Conservative Vector Field in Space Suppose that M, N, and P have continuous first partial derivatives in an open sphere Q in space. The vector field given by F(x, y, z) = Mi + Nj + Pk is conservative if and only if Curl F(x, y, z) = 0 That is, F is conservative if and only if:

  13. 11. Determine whether the vector field is conservative. If it is, find a potential function for the vector field(Similar to p.1068 #57-62)Hint: F(x, y) = fxi + fyj+ fzkafter you get fx, fy, and fz, take the integral with respect to the variable to get the potential function

  14. Definition of Divergence of a Vector Field The divergence of F(x, y) = Mi + Nj is: The divergence of F(x, y, z) = Mi + Nj + Pk is: If div F = 0, then F is said to be divergence free

  15. 12. Find the divergence of the vector field F(Similar to p.1068 #63-66)

  16. 13. Find the divergence of the vector field F(Similar to p.1068 #63-66)

  17. 14. Find the divergence of the vector field Fat the given point(Similar to p.1068 #63-66)

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