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Chapter 16

Chapter 16. Section 16.9 Reconstructing a Function from its Gradient. Potential Functions and Antigradients

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Chapter 16

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  1. Chapter 16 Section 16.9 Reconstructing a Function from its Gradient

  2. Potential Functions and Antigradients Given a vector field F if there exists a function (the function might not exist !) such that the gradient of is the vector field F, then the function is called the antigradient (like antiderivative) for F, or is the potential function for the vector field F. The function is the antigradient or potential function for the vector field F. Has an antigradient if: The condition for a vector field of 2 independent variables to have an antigradient is for the partial derivatives with respect to the opposite variables to be equal. Determine if satisfies . This will mean that Integrate with respect to x: Derivative with respect to y: Set equal to : Cancel x part to solve for : Integrate with respect to y to find : Substitute to find z: Finding the Antigradient The process of determining if a vector field F has an antigradient has 7 steps.

  3. Example Determine if the vector field has an antigradient and find it if so. 1. Check partial derivatives: 2. Integrate treating x as the variable: 3. Take derivative treating y as the variable: 4. Set equal to N: 5. Eliminate the x part: 6. Integrate with respect to y variable: 7. Substitute into z to get antigradient:

  4. Example Determine if has an antigradient and find it does. 1. Check partial derivatives: 2. Integrate treating x as the variable: 3. Take derivative treating y as the variable: 4. Set equal to N: 5. Eliminate the x part: 6. Integrate with respect to y variable: 7. Substitute into z to get antigradient:

  5. Example Determine if the vector field of 3 variables has an antigradient and find it does. 1. Check pair for equal partials: 2. Integrate out x part: 7. Substitute back in w: 8. Derivative with respect to z: 3. Derivative with respect to y: 9. Set equal to P: 4. Set equal to N: 10. Cancel out x and y parts: 5. Cancel out x part: 11. Integrate with respect to z: 6. Integrate with respect to y: 12. Substitute back in w:

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