Vector fields
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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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Vector fields

Vector Fields


1 computer f and sketch several representative vectors in the vector field similar to p 1067 7 16

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


Vector fields

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


Vector fields

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


Vector fields

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


Vector fields

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


Vector fields

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


Vector fields

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


Vector fields

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


Vector fields

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


10 find curl f for the vector field at the given point similar to p 1067 49 52

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


Test for conservative vector field in space

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:


Vector fields

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


Definition of divergence of a vector field

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


12 find the divergence of the vector field f similar to p 1068 63 66

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


13 find the divergence of the vector field f similar to p 1068 63 66

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


14 find the divergence of the vector field f at the given point similar to p 1068 63 66

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


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