Question

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

Short answer

Question: Prove the identity involving the divergence, curl, and scalar and vector products of two vector fields \(\mathbf{F}\) and \(\mathbf{G}\): $$\nabla \cdot(\mathbf{F} \times \mathbf{G})=\mathbf{G} \cdot(\nabla \times \mathbf{F})-\mathbf{F} \cdot(\nabla \times \mathbf{G}).$$ Answer: Following a series of calculations, we found that the expressions for the divergence of the cross product of \(\mathbf{F}\) and \(\mathbf{G}\) and the subtraction of the dot products of \(\mathbf{G}\) and \(\mathbf{F}\) with their respective curls are the same. Therefore, we have proved the identity: $$\nabla \cdot(\mathbf{F} \times \mathbf{G})=\mathbf{G} \cdot(\nabla \times \mathbf{F})-\mathbf{F} \cdot(\nabla \times \mathbf{G}).$$

Step-by-step solution

Define the given terms and identity

The given terms are scalar-valued function \(\varphi\), and differentiable vector fields \(\mathbf{F}\) and \(\mathbf{G}\). The identity we want to prove is: $$\nabla \cdot(\mathbf{F} \times \mathbf{G})=\mathbf{G} \cdot(\nabla \times \mathbf{F})-\mathbf{F} \cdot(\nabla \times \mathbf{G}).$$

Write the components of the given vector fields

Let's write the components of the vector fields \(\mathbf{F}\) and \(\mathbf{G}\), it will help us in the subsequent steps. \(\mathbf{F} = (F_{1}, F_{2}, F_{3})\) and \(\mathbf{G} = (G_{1}, G_{2}, G_{3}).\)

Compute the cross product between \(\mathbf{F}\) and \(\mathbf{G}\)

We will compute the cross product of \(\mathbf{F}\) and \(\mathbf{G}\) using their components: $$(\mathbf{F} \times \mathbf{G})=\begin{vmatrix}\mathbf{i}&\mathbf{j}&\mathbf{k}\\F_{1}&F_{2}&F_{3}\\G_{1}&G_{2}&G_{3}\end{vmatrix}=\mathbf{i}(F_2G_3 - F_3G_2) - \mathbf{j}(F_1G_3 - F_3G_1) + \mathbf{k}(F_1G_2 - F_2G_1).$$

Calculate the divergence of the cross product

Now, let's calculate the divergence of our cross product: $$\nabla \cdot (\mathbf{F} \times \mathbf{G}) = \frac{\partial}{\partial x}(F_2G_3 - F_3G_2) - \frac{\partial}{\partial y}(F_1G_3 - F_3G_1) + \frac{\partial}{\partial z}(F_1G_2 - F_2G_1).$$

Calculate the curls of \(\mathbf{F}\) and \(\mathbf{G}\)

Let's now calculate the curl of \(\mathbf{F}\) and \(\mathbf{G}\): $$\nabla \times \mathbf{F} = \begin{vmatrix}\mathbf{i}&\mathbf{j}&\mathbf{k}\\\frac{\partial}{\partial x}&\frac{\partial}{\partial y}&\frac{\partial}{\partial z}\\F_1&F_2&F_3\end{vmatrix}$$ and $$\nabla \times \mathbf{G} = \begin{vmatrix}\mathbf{i}&\mathbf{j}&\mathbf{k}\\\frac{\partial}{\partial x}&\frac{\partial}{\partial y}&\frac{\partial}{\partial z}\\G_1&G_2&G_3\end{vmatrix}.$$

Calculate the dot products of \(\mathbf{G}\) and \(\mathbf{F}\) with the curls

Now, we need to calculate the dot product of \(\mathbf{G}\) with \(\nabla \times \mathbf{F}\) and the dot product of \(\mathbf{F}\) with \(\nabla \times \mathbf{G}\). After calculating the dot products, we can subtract the two results: $$\mathbf{G} \cdot(\nabla \times \mathbf{F})-\mathbf{F} \cdot(\nabla \times \mathbf{G}) = (\mathbf{G} \cdot \nabla)\mathbf{F} - (\mathbf{F} \cdot \nabla)\mathbf{G} \quad (1).$$

Prove the identity

Now, we will compare the results of Steps 4 and 6. If they are the same, then the identity is true. Let's simplify the result in Step 6: $$\mathbf{G} \cdot(\nabla \times \mathbf{F})-\mathbf{F} \cdot(\nabla \times \mathbf{G}) = \frac{\partial}{\partial x}(F_2G_3 - F_3G_2) - \frac{\partial}{\partial y}(F_1G_3 - F_3G_1) + \frac{\partial}{\partial z}(F_1G_2 - F_2G_1).$$ Comparing the results from Steps 4 and 6, we see that they are indeed the same. Therefore, the identity is proved: $$\nabla \cdot(\mathbf{F} \times \mathbf{G})=\mathbf{G} \cdot(\nabla \times \mathbf{F})-\mathbf{F} \cdot(\nabla \times \mathbf{G}).$$

Key concepts

Divergence of Cross Product

Understanding the divergence of a cross product is a fundamental concept in vector calculus. The divergence operator, denoted by abla \cdot\text{, measures the rate at which a vector field spreads out or diverges from a given point. When applied to the cross product of two vector fields, abla \cdot(\mathbf{F} \times \mathbf{G}), we tap into the interaction between these fields.

For the vector fields \(\mathbf{F} = (F_{1}, F_{2}, F_{3})\) and \(\mathbf{G} = (G_{1}, G_{2}, G_{3})\), their cross product produces a new vector field whose divergence can give insight into the rotational effects and shear within the fields. This leads to the identity we are proving: \[abla \cdot(\mathbf{F} \times \mathbf{G}) = \mathbf{G} \cdot(abla \times \mathbf{F}) - \mathbf{F} \cdot(abla \times \mathbf{G})\]
To appreciate the significance of this identity, it's vital to acknowledge its implications in physical applications such as fluid dynamics and electromagnetism, where it is often used to simplify the computations involving rotational and divergence effects.

Curl of a Vector Field

The curl of a vector field, denoted by abla \times\text{, measures the tendency of the field to rotate around a point. When you calculate the curl of a differentiable vector field, such as \(\mathbf{F}\) and \(\mathbf{G}\), you're looking at a local spinning effect of the field.

By using the determinant of a matrix that includes the unit vectors \(\mathbf{i}, \mathbf{j}, \mathbf{k}\) and the partial derivatives of the vector field components, you can extract the circulation or swirling motion. The resulting vector indicates the axis of rotation and the amount of 'twist' at a point. Physically, this is essential for understanding phenomena like the rotation of fluid elements in a flow or the magnetic field in the presence of electric currents.

Vector Calculus

Vector calculus is a branch of mathematics focused on differentiation and integration of vector fields. This encompasses operations like the gradient, divergence, and curl, among others.

A vector field is a function that assigns a vector to each point in space. For example, in physics, a vector field might represent the velocity of moving fluid at every point in the space occupied by the fluid. Vector calculus provides us with the tools to analyze this space and understand the behavior of fields under various conditions. Key applications include examining electric and magnetic fields, fluid flow, and forces in mechanics.

Scalar-valued Function

In contrast to vector fields, a scalar-valued function assigns a single numerical value to each point in a space. The function \(\varphi\) mentioned in the problem is an example of a scalar-valued function.

Scalar fields are pervasive in physics and engineering. For example, temperature distribution in a room can be modeled as a scalar field, where to each point in the room's volume, there is a numerical temperature value associated. In calculus, finding the gradient of a scalar-valued function gives a vector field that points in the direction of the greatest rate of increase of the function.

Differentiable Vector Fields

Differentiable vector fields are fields where each component function has continuous partial derivatives. This differentiable property is crucial when applying vector calculus operations such as divergence or curl.

A differentiable vector field like \(\mathbf{F}\) or \(\mathbf{G}\) in our exercise ensures that we can perform computations smoothly and meaningfully interpret the results. The existence of derivatives at every point ensures we can analyze the behavior of the vector field's magnitude and direction with respect to small changes in the surrounding space—key for studying the dynamic behavior of systems in applied sciences and engineering.