Question

Find the divergence of the following vector fields. $$\mathbf{F}=\langle 2 x, 4 y,-3 z\rangle$$

   Find the divergence of the following vector fields.
$$\mathbf{F}=\langle 2 x, 4 y,-3 z\rangle$$

Calculus for Scientists and Engineers: Early Transcendental

William Briggs, Lyle… 1st Edition

Chapter 15, Problem 9

Instant Answer

Solved by Expert Subhadeepta Sahoo

08/19/2020

Step 1

Step 1: The divergence of a vector field $\mathbf{F}=\langle F_1, F_2, F_3\rangle$ in Cartesian coordinates is given by the formula: $$\nabla \cdot \mathbf{F} = \frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y} + \frac{\partial F_3}{\partial z}$$ Show more…

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Find the divergence of the following vector fields. $$\mathbf{F}=\langle 2 x, 4 y,-3 z\rangle$$

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Key Concepts

Divergence

Divergence is an operator in vector calculus that measures the magnitude of a field's source or sink at a given point. It is defined as the sum of the partial derivatives of the vector field's components with respect to their respective independent variables, essentially quantifying how much the vector field 'diverges' from a point.

Partial Derivative

Partial derivatives are a fundamental concept in multivariable calculus, used to measure the rate at which a function changes with respect to one of several variables while holding the others constant. They are instrumental in computing operators like divergence, which require differentiating each component of a vector field with respect to its corresponding variable.

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