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(c) Vf(x, y, z) = (2xy, x² – 2yz, 4 - y²) -

(c) Vf(x, y, z) = (2xy, x² – 2yz, 4 - y²) -

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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I just need help with part C.

**Problem 13: [Anti-gradients]** Find the general anti-gradient, \( f \), i.e. \( f \) was the original function, and you are given the gradient of the function. (a) \(\nabla f(x, y) = \langle 6x + 4y^3, -2 + 12xy^2 \rangle\) (b) \(\nabla f(x, y) = \left\langle \frac{6x}{y^2} - 2e^{-2x}, -\frac{6x^2}{y^3} + 4y \right\rangle\) (c) \(\nabla f(x, y, z) = \langle 2xy, x^2 - 2yz, 4 - y^2 \rangle\) The task involves finding the original function \( f \) from the given gradients by integrating each component with respect to its corresponding variable.
Transcribed Image Text:**Problem 13: [Anti-gradients]** Find the general anti-gradient, \( f \), i.e. \( f \) was the original function, and you are given the gradient of the function. (a) \(\nabla f(x, y) = \langle 6x + 4y^3, -2 + 12xy^2 \rangle\) (b) \(\nabla f(x, y) = \left\langle \frac{6x}{y^2} - 2e^{-2x}, -\frac{6x^2}{y^3} + 4y \right\rangle\) (c) \(\nabla f(x, y, z) = \langle 2xy, x^2 - 2yz, 4 - y^2 \rangle\) The task involves finding the original function \( f \) from the given gradients by integrating each component with respect to its corresponding variable.
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