In this worksheet, we will practice examining gradient vector fields produced from a multivariate function.

**Q1: **

Compute the gradient of the function .

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**Q2: **

Find the gradient of .

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**Q3: **

Find the gradient of

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**Q4: **

Compute the gradient of the function .

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**Q5: **

Find the gradient of .

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**Q6: **

Compute the gradient for

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**Q7: **

Compute the gradient for

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**Q8: **

Compute the gradient for

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**Q9: **

Compute the gradient for

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**Q10: **

Find a function so that the vector field is a gradient field.

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**Q11: **

We explore an example where a vector field satisfies but does not come from a potential. On the plane with the origin removed, consider the vector field .

On the (open) half-plane , we can define the angle function . This is well defined and gives a value between and . What is the gradient ?

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Using the figure shown, use above to define the angle function on the region by a suitable composition with a rotation.

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What is ?

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Since and agree on the quadrant , , we can define the angle at each point of the union with values between and . Using this, what is , where is the arc of the unit circle from to ? Answer in terms of .

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In the same way, we can define on the half-plane and on . Hence, evaluate the line integral around the circle of radius , starting from and going counterclockwise back to , stating your answer in terms of .

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**Q12: **

For in Cartesian coordinates, find in cylindrical coordinates.

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**Q13: **

Suppose with . Express the gradient (viewed as a matrix) in terms of the matrix , where , and a matrix of partial derivatives of .

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**Q14: **

Suppose with and that for a point . Express the gradient (viewed as a matrix) in terms of the matrix and a matrix of partial derivatives of .

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