Let 𝐅 be a differentiable vector field, and let g(x, y, z)...

Let $\mathbf{F}$ be a differentiable vector field, and let $g(x, y, z)$ be a differentiable scalar function. Verify the following identities. a. $\nabla \cdot(g \mathbf{F})=g \nabla \cdot \mathbf{F}+\nabla g \cdot \mathbf{F}$ b. $\nabla \times(g \mathbf{F})=g \nabla \times \mathbf{F}+\nabla g \times \mathbf{F}$

Let $\mathbf{F}$ be a differentiable vector field, and let $g(x, y, z)$ be a differentiable scalar function. Verify the following identities.
a. $\nabla \cdot(g \mathbf{F})=g \nabla \cdot \mathbf{F}+\nabla g \cdot \mathbf{F}$
b. $\nabla \times(g \mathbf{F})=g \nabla \times \mathbf{F}+\nabla g \times \mathbf{F}$

Key Concepts

Product Rule in Vector Calculus

The product rule in vector calculus generalizes the familiar product rule from single-variable calculus to operations involving scalar and vector fields. When a scalar function multiplies a vector field, the gradient, divergence, and curl of the product are expressed as sums of terms involving derivatives of both functions. This rule is essential for understanding how differential operators act on combinations of functions.

Divergence

Divergence is a scalar measure of a vector field’s tendency to originate from or converge into a point. In the context of vector calculus, it is given by the dot product of the del operator with the vector field. When the divergence is applied to a product of a scalar function and a vector field, the product rule helps to distribute the differentiation, leading to one term involving the divergence of the vector field and another term involving the gradient of the scalar function dotted with the vector field.

Curl

Curl is a vector operator that measures the rotation of a vector field around a point. It is defined as the cross product of the del operator with the vector field. Applying the curl to a product of a scalar function and a vector field also requires a product rule, resulting in one term that multiplies the scalar function with the curl of the vector field and a second term that involves the cross product of the gradient of the scalar function with the vector field.

Gradient

The gradient of a scalar function is a vector that points in the direction of the greatest rate of increase of the function, with its magnitude being the rate of increase. It is instrumental in formulating the product rules for divergence and curl when a scalar function is multiplied with a vector field, allowing these derivative operations to be decomposed into simpler components.

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