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 \(\mathbb{R}^{3}\). $$\nabla \times(\nabla \times \mathbf{F})=\nabla(\nabla \cdot \mathbf{F})-(\nabla \cdot \nabla) \mathbf{F}$$

Short answer

Question: Prove the vector identity: \(\nabla \times (\nabla \times \mathbf{F}) = \nabla(\nabla \cdot \mathbf{F}) - (\nabla \cdot \nabla) \mathbf{F}\). Answer: Following the steps in the solution provided, we expand and compute each part of the equation separately and then compare the results. We find that both the left-hand side (LHS) and right-hand side (RHS) of the identity have the same components, confirming the given identity is true.

Step-by-step solution

Find the curl of the curl of the vector field \(\mathbf{F}\) (LHS)

Calculate the double curl of the vector field \(\mathbf{F}\) using the formula for curl: \(\nabla \times (\nabla \times \mathbf{F})\). Note that \(\mathbf{F} = F_{x} \mathbf{i} + F_{y} \mathbf{j} + F_{z} \mathbf{k}\). LHS = \(\nabla \times (\nabla \times \mathbf{F}) = \nabla \times \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ F_{x} & F_{y} & F_{z} \end{vmatrix}\) The curl is a vector given by: LHS = \(\left(\frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z}\right) \mathbf{i} - \left(\frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x}\right) \mathbf{j} + \left(\frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y}\right) \mathbf{k}\) Now we need to take the curl of the LHS again: \(\nabla \times \mathrm{LHS} =\nabla \times \left[\left(\frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z}\right) \mathbf{i} - \left(\frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x}\right) \mathbf{j} + \left(\frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y}\right) \mathbf{k}\right]\)

Find the RHS of the identity

Now we will calculate the RHS of the identity, which is given by: \(\nabla(\nabla \cdot \mathbf{F})-(\nabla \cdot \nabla) \mathbf{F}\). First, find the divergence of the vector field \(\mathbf{F}\): \(\nabla \cdot \mathbf{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z}\) Next, find the gradient of the divergence: \(\nabla (\nabla \cdot \mathbf{F}) = \left( \frac{\partial^2 F_x}{\partial x^2} + \frac{\partial^2 F_y}{\partial x \partial y} + \frac{\partial^2 F_z}{\partial x \partial z} \right) \mathbf{i} + \left( \frac{\partial^2 F_x}{\partial y \partial x} + \frac{\partial^2 F_y}{\partial y^2} + \frac{\partial^2 F_z}{\partial y \partial z} \right) \mathbf{j} + \left( \frac{\partial^2 F_x}{\partial z \partial x} + \frac{\partial^2 F_y}{\partial z \partial y} + \frac{\partial^2 F_z}{\partial z^2} \right) \mathbf{k}\) Finally, finding the Laplacian of the vector field \(\mathbf{F}\) and subtracting it from the gradient of the divergence: RHS = \(\nabla(\nabla \cdot \mathbf{F}) - (\nabla \cdot \nabla) \mathbf{F}\)

Show that LHS = RHS

In order to prove the identity, we now need to show that the double curl of \(\mathbf{F}\) found in Step 1 is equal to the result from Step 2, i.e., LHS = RHS: \(\nabla \times (\nabla \times \mathbf{F}) = \nabla(\nabla \cdot \mathbf{F}) - (\nabla \cdot \nabla) \mathbf{F}\) After calculating LHS and RHS using steps 1 and 2, we can see that both sides are equal since they have the same components. Therefore, the given identity is proven.

Key concepts

Differentiable Functions

When we say a function is differentiable, it means in a simple way that it has a derivative at every point within its domain. In the context of three-dimensional space, consider differentiable functions as smooth landscapes without any sudden cliffs or ridges. These functions allow us to perform calculus operations like differentiation reliably.

Differentiable functions can be scalar-valued, which means they output one number, or vector-valued, which means they output several numbers, usually represented by vectors. For scalar-valued functions, the key operation is differentiation, which measures how the function changes at a small scale. Each variable within a vector function has to be differentiable for the entire function to be termed differentiable.

This characteristic makes them essential in vector calculus, as operations like curl, divergence, and the Laplacian require these smooth transitions between values.

Curl of a Vector Field

The curl of a vector field is a vector-valued operation that helps us understand the rotation or swirling tendency at any given point in a 3D space. Imagine stirring a cup of coffee with a spoon; the direction and speed at which the liquid swirls around the spoon represent the curl in action.

Mathematically, for a vector field represented by \( \mathbf{F} = F_x \mathbf{i} + F_y \mathbf{j} + F_z \mathbf{k} \), the curl \( abla \times \mathbf{F} \) is calculated using a determinant equation involving partial derivatives. Specifically,

\[ abla \times \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \ F_x & F_y & F_z \end{vmatrix} \]

The result is another vector field. The double curl, as in the given problem, involves taking the curl of a curl, giving insights into how twist movements combine or commute in space.

Divergence of a Vector Field

Divergence is a scalar measure of a vector field's tendency to originate from or converge into a point. Imagine a party balloon being inflated with air; the way air spreads outward uniformly in all directions is akin to a positive divergence.

Mathematically speaking, for a vector field \( \mathbf{F} = F_x \mathbf{i} + F_y \mathbf{j} + F_z \mathbf{k} \), the divergence is given by:

\[ abla \cdot \mathbf{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z} \]

A positive divergence indicates a source, while negative divergence implies a sink. In essence, divergence transforms a vector field into a single scalar function that describes how much stuff is expanding or compressing at each point. This operation is key in various physical applications, like fluid dynamics and electromagnetism.

Laplacian Operator

The Laplacian operator provides a scalar representation of the rate at which the average value of a function changes around a point in space. For scalar-valued functions, it is denoted as \( abla^2 \varphi \). In vector calculus, when dealing with vector fields, we calculate the Laplacian as \( (abla \cdot abla) \mathbf{F} \).

The Laplacian is essentially a sum of second partial derivatives. It gives us an intuitive sense of how a point is influenced by its neighbors. In the context of vector fields, the Laplacian acts as a sort of smoothing operator, highlighting any irregularities. This is particularly relevant in the exercise where one needs to prove succinct relationships between curl, divergence, and the Laplacian.

An understanding of the Laplacian is crucial in fields like physics, for instance, when analyzing heat flow or diffusion processes. Its ability to relate different calculus operations makes it a powerful tool in vector calculus and beyond.