Question

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

Short answer

Question: Prove the following identity: $\nabla \cdot(\varphi \mathbf{F})=\nabla \varphi \cdot \mathbf{F}+\varphi \nabla \cdot \mathbf{F}$, where $\varphi$ is a differentiable scalar-valued function and $\mathbf{F}$ is a differentiable vector field. Answer: The proof of the identity can be found in the step-by-step solution above, which demonstrates that $\nabla \cdot(\varphi \mathbf{F})=\nabla \varphi \cdot \mathbf{F}+\varphi \nabla \cdot \mathbf{F}$ by calculating the gradient of the scalar-valued function $\varphi$, the scalar product of the gradient of $\varphi$ with the vector field $\mathbf{F}$, the product of $\varphi$ and $\mathbf{F}$, the divergence of the product of $\varphi$ and $\mathbf{F}$, and finally confirming the given identity.

Step-by-step solution

Determine the gradient of scalar-valued function \(\varphi\)

We will begin by determining the gradient of \(\varphi\). The gradient of a scalar-valued function is a vector whose components are partial derivatives of the function with respect to each of the Cartesian coordinates \(x\), \(y\), and \(z\): $$\nabla \varphi = \left( \frac{\partial \varphi}{\partial x}, \frac{\partial \varphi}{\partial y}, \frac{\partial \varphi}{\partial z} \right)$$

Determine the scalar product of the gradient of \(\varphi\) with the vector field \(\mathbf{F}\)

Next, we will find the scalar product of \(\nabla \varphi\) and \(\mathbf{F}\). Suppose \(\mathbf{F} = (F_x, F_y, F_z)\), then the scalar product is: $$\nabla \varphi \cdot \mathbf{F} = \frac{\partial \varphi}{\partial x} F_x + \frac{\partial \varphi}{\partial y} F_y + \frac{\partial \varphi}{\partial z} F_z$$

Determine the product of \(\varphi\) and the vector field \(\mathbf{F}\)

Now, we will compute the product of \(\varphi\) and \(\mathbf{F}\). The product of a scalar-valued function with a vector field is a new vector field whose components are the product of the scalar function and each component of the original vector field: $$\varphi \mathbf{F} = ( \varphi F_x, \varphi F_y, \varphi F_z)$$

Determine the divergence of the product of \(\varphi\) and \(\mathbf{F}\)

We will calculate the divergence of the product obtained in the previous step. The divergence of a vector field is the sum of the partial derivatives of its components with respect to the Cartesian coordinates \(x\), \(y\), and \(z\): $$\nabla \cdot(\varphi \mathbf{F}) = \frac{\partial (\varphi F_x)}{\partial x} + \frac{\partial (\varphi F_y)}{\partial y} + \frac{\partial (\varphi F_z)}{\partial z}$$ Using the product rule for derivatives, we get: $$\nabla \cdot(\varphi \mathbf{F}) = \frac{\partial \varphi}{\partial x} F_x + \frac{\partial F_x}{\partial x}\varphi + \frac{\partial \varphi}{\partial y} F_y + \frac{\partial F_y}{\partial y}\varphi + \frac{\partial \varphi}{\partial z} F_z + \frac{\partial F_z}{\partial z}\varphi$$ Combining the terms, we can rewrite the above expression as: $$\nabla \cdot(\varphi \mathbf{F}) = \nabla \varphi \cdot \mathbf{F} + \varphi (\frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z})$$

Conclude the proof

We notice that: $$\nabla \cdot \mathbf{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z}$$ Thus, we can substitute this into our previous expression: $$\nabla \cdot(\varphi \mathbf{F}) = \nabla \varphi \cdot \mathbf{F} + \varphi (\nabla \cdot \mathbf{F})$$ This confirms the given identity: $$\nabla \cdot(\varphi \mathbf{F})=\nabla \varphi \cdot \mathbf{F}+\varphi \nabla \cdot \mathbf{F}$$

Key concepts

Gradient of Scalar Field

The gradient of a scalar field is a fundamental concept in vector calculus, representing the directional change of a scalar-valued function. Imagine you're on a mountain, and the height of the mountain at any point represents the value of your scalar function. The gradient at any point shows the direction of the steepest ascent and its magnitude indicates how steep the climb is.

In mathematical terms, for a function \( \varphi \) defined in three dimensions, the gradient is a vector composed of partial derivatives along each axis:\[abla \varphi = \left( \frac{\partial \varphi}{\partial x}, \frac{\partial \varphi}{\partial y}, \frac{\partial \varphi}{\partial z} \right)\]
Each component of this gradient vector points in the direction where the function increases most rapidly. This concept lays the foundation for understanding other identities in vector calculus.

Divergence of a Vector Field

Divergence is another core concept that measures the magnitude of a vector field's source or sink at a given point. In a physical sense, if you consider air flowing around a room, the divergence at any point tells you whether that point is acting as a source, with air flowing out, or a sink, with air flowing in.

The divergence is mathematically defined as the sum of the partial derivatives of a vector field's components:\[abla \cdot \mathbf{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z}\]
Thus, a positive divergence indicates a 'source' of the vector field, while a negative divergence indicates a 'sink'. Zero divergence means the vector field is neither expanding nor contracting.

Scalar and Vector Product

In vector calculus, we often use two types of products: the scalar (dot) product and the vector (cross) product. The scalar product of two vectors results in a scalar quantity, essentially giving a measure of their directional alignment. For instance, if you're pulling on a rope, the scalar product tells you how much of your force is acting in the direction of the rope.

For \( abla \varphi \) and \( \mathbf{F} \) as defined earlier, the scalar product is computed as:\[abla \varphi \cdot \mathbf{F} = \frac{\partial \varphi}{\partial x} F_x + \frac{\partial \varphi}{\partial y} F_y + \frac{\partial \varphi}{\partial z} F_z\]
The vector product, on the other hand, results in a new vector that is orthogonal to both original vectors and has a magnitude representative of the parallelogram spanned by them.

Product Rule for Derivatives

The product rule is a derivative rule that applies when you are differentiating the product of two functions. In simpler terms, if you're working on a problem involving the rate of change of two quantities that depend on each other, the product rule allows you to find the overall rate of change.

In vector calculus, we extend this concept as follows:\[abla \cdot(\varphi \mathbf{F}) = \frac{\partial (\varphi F_x)}{\partial x} + \frac{\partial (\varphi F_y)}{\partial y} + \frac{\partial (\varphi F_z)}{\partial z}\]
By using the product rule, we split these derivatives, which allows us to relate the divergence of a vector field multiplied by a scalar to the gradient of the scalar function and the divergence of the original vector field.

Vector Calculus Proofs

Vector calculus proofs are rigorous arguments that verify the truths of propositions or identities involving vector functions and operators. Similar to proof in traditional calculus, vector calculus proofs involve logical reasoning and often use established rules.

The previous sections described the individual components necessary to understand the proof of the vector calculus identity involving the gradient, divergence, and product rule. Bringing those elements together, we can conclusively show the relationship known as the Product Rule for the divergence of a product of a scalar field and a vector field. These types of proofs are crucial for establishing the foundational relationships used throughout physics and engineering disciplines.