Question

Prove the following identities. Assume that $\phi$ is $a$ differentiable scalar-valued function and $\mathbf{F}$ and $\mathbf{G}$ are differentiable vector fields, all defined on a region of ${\mathbb{R}}^3$. $$\mathrm{\nabla }\cdot (\phi \mathbf{F})=\mathrm{\nabla }\phi \cdot \mathbf{F}+\phi \mathrm{\nabla }\cdot \mathbf{F}\text{ (Product Rule) }$$

Short answer

Answer: The divergence of the product of a scalar-valued function $\phi$ and a vector field $\mathbf{F}$ is equal to the gradient of the scalar function dotted with the vector field plus the scalar function times the divergence of the vector field, given by the following identity: $\mathrm{\nabla }\cdot (\phi \mathbf{F})=\mathrm{\nabla }\phi \cdot \mathbf{F}+\phi \mathrm{\nabla }\cdot \mathbf{F}$.

Step-by-step solution

Recall the definitions of gradient, divergence, and dot product.

The gradient of a scalar function $\phi$ is denoted by $\mathrm{\nabla }\phi$ and is a vector with the partial derivatives of the scalar function as its components: $$\mathrm{\nabla }\phi =(\frac{\partial \phi }{\partial x},\frac{\partial \phi }{\partial y},\frac{\partial \phi }{\partial z}).$$ The divergence of a vector field $\mathbf{F}$ is denoted by $\mathrm{\nabla }\cdot \mathbf{F}$ and is the sum of the partial derivatives of the components of the vector field: $$\mathrm{\nabla }\cdot \mathbf{F}=\frac{\partial F_x}{\partial x}+\frac{\partial F_y}{\partial y}+\frac{\partial F_z}{\partial z}.$$ A dot product between two vectors, $\mathbf{A}$ and $\mathbf{B}$, is defined as the sum of the products of their corresponding components: $$\mathbf{A}\cdot \mathbf{B}=A_x{B}_x+A_y{B}_y+A_z{B}_z.$$

Compute the divergence of the scalar-valued function times the vector field.

We have $\mathrm{\nabla }\cdot (\phi \mathbf{F})=\mathrm{\nabla }\cdot (\phi F_x,\phi F_y,\phi F_z)$. By applying the definition of divergence, we have: $$\mathrm{\nabla }\cdot (\phi \mathbf{F})=\frac{\partial (\phi F_x)}{\partial x}+\frac{\partial (\phi F_y)}{\partial y}+\frac{\partial (\phi F_z)}{\partial z}.$$

Apply the product rule for partial derivatives.

For each of the partial derivatives in the previous expression, apply the product rule for partial derivatives: $$\frac{\partial (\phi F_x)}{\partial x}=\frac{\partial \phi }{\partial x}{F}_x+\phi \frac{\partial F_x}{\partial x},$$ $$\frac{\partial (\phi F_y)}{\partial y}=\frac{\partial \phi }{\partial y}{F}_y+\phi \frac{\partial F_y}{\partial y},$$ $$\frac{\partial (\phi F_z)}{\partial z}=\frac{\partial \phi }{\partial z}{F}_z+\phi \frac{\partial F_z}{\partial z}.$$

Substitute the expressions from Step 3 back into the expression for the divergence.

We have: $$\mathrm{\nabla }\cdot (\phi \mathbf{F})=(\frac{\partial \phi }{\partial x}{F}_x+\phi \frac{\partial F_x}{\partial x})+(\frac{\partial \phi }{\partial y}{F}_y+\phi \frac{\partial F_y}{\partial y})+(\frac{\partial \phi }{\partial z}{F}_z+\phi \frac{\partial F_z}{\partial z}).$$

Rewrite the expression as a sum of two terms.

Notice the terms with $\frac{\partial \phi }{\partial x}{F}_x,\frac{\partial \phi }{\partial y}{F}_y,\frac{\partial \phi }{\partial z}{F}_z$ are a dot product and the terms $\phi \frac{\partial F_x}{\partial x},\phi \frac{\partial F_y}{\partial y},\phi \frac{\partial F_z}{\partial z}$ are a product: $$\mathrm{\nabla }\cdot (\phi \mathbf{F})=(\frac{\partial \phi }{\partial x}{F}_x+\frac{\partial \phi }{\partial y}{F}_y+\frac{\partial \phi }{\partial z}{F}_z)+(\phi \frac{\partial F_x}{\partial x}+\phi \frac{\partial F_y}{\partial y}+\phi \frac{\partial F_z}{\partial z}).$$

Replace the dot product and product from Step 5 with their respective symbols.

Now we have: $$\mathrm{\nabla }\cdot (\phi \mathbf{F})=\mathrm{\nabla }\phi \cdot \mathbf{F}+\phi (\frac{\partial F_x}{\partial x}+\frac{\partial F_y}{\partial y}+\frac{\partial F_z}{\partial z}).$$

Recognize the divergence of the vector field.

Notice the expression in parentheses is the definition of the divergence of $\mathbf{F}$: $$\mathrm{\nabla }\cdot (\phi \mathbf{F})=\mathrm{\nabla }\phi \cdot \mathbf{F}+\phi \mathrm{\nabla }\cdot \mathbf{F}.$$ This proves the identity as desired.

Key concepts

Product Rule

The product rule is a fundamental concept in calculus. It describes how to differentiate a product of two functions. In the context of vector calculus, it's crucial when working with systems involving scalar fields and vector fields.
For a product of a differentiable scalar field $\phi$ and a vector field $\mathbf{F}$, the product rule enables us to separate the derivative of the product into manageable parts. According to the product rule in vector calculus, the divergence of a product of a scalar function and a vector field can be expressed as:
$$abla\cdot (\phi \mathbf{F})=abla\phi \cdot \mathbf{F}+\phi abla\cdot \mathbf{F}$$
This equation tells us that the divergence of the product is the divergence of the vector field $\mathbf{F}$ scaled by the scalar field $\phi$ added to the dot product of the gradient of the scalar field and the vector field itself. The product rule simplifies handling complex expressions by breaking them down into individual derivatives.

Divergence

Divergence is a vector operation that measures the magnitude of a field's source or sink at a given point. For a vector field $\mathbf{F}=(F_x,F_y,F_z)$, the divergence is a scalar value computed as:
$$abla\cdot \mathbf{F}=\frac{\partial F_x}{\partial x}+\frac{\partial F_y}{\partial y}+\frac{\partial F_z}{\partial z}$$
This formula signifies how much the vector field is 'spreading out' or 'converging' at any point in space.

The divergence is an essential concept in fluid dynamics, electromagnetism, and other fields, where it represents, for example, the rate at which a fluid is expanding or contracting. In the product rule identity, divergence plays a vital role in ensuring that the calculations accurately reflect how the scalar field $\phi$ affects the behavior of the vector field $\mathbf{F}$.
Understanding divergence allows us to comprehend complex topics like the continuity equation and Gauss's theorem, as it provides insight into the inward or outward flux from a point.

Gradient

The gradient of a scalar field is a vector field that points in the direction of the steepest increase of the function. For a scalar function $\phi$, its gradient is calculated as:
$$abla\phi =(\frac{\partial \phi }{\partial x},\frac{\partial \phi }{\partial y},\frac{\partial \phi }{\partial z})$$

This vector field helps to understand how $\phi$ changes in space. The length of the gradient vector at any point gives the rate of change of $\phi$ at that point.

In our product rule identity:
$$abla\phi \cdot \mathbf{F}$$
This term represents how $\phi$ influences the direction and magnitude of the vector field $\mathbf{F}$. The gradient is crucial for expressing various physical phenomena, from heat conduction (described by Fourier's Law) to better understanding potential fields in physics such as gravity or electromagnetism.

Dot Product

The dot product is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. For two vectors $\mathbf{A}=(A_x,A_y,A_z)$ and $\mathbf{B}=(B_x,B_y,B_z)$, it's computed as:
$$\mathbf{A}\cdot \mathbf{B}=A_x{B}_x+A_y{B}_y+A_z{B}_z$$

The dot product gives a measure of how parallel two vectors are. It's zero if the vectors are perpendicular and positive if they point in roughly the same direction.

In the product rule identity, the dot product $abla\phi \cdot \mathbf{F}$
is essential because it combines the directional change of the scalar field $\phi$ with the orientation of the vector field $\mathbf{F}$. Understanding this concept helps in various applications, such as determining work done by a force or analyzing projections of vectors in physics and engineering.