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

Short answer

Question: Prove that for any vector fields $\mathbf{F}$ and $\mathbf{G}$, the following identity holds: $$\nabla \cdot(\mathbf{F} \times \mathbf{G})=\mathbf{G} \cdot(\nabla \times \mathbf{F})-\mathbf{F} \cdot(\nabla \times \mathbf{G})$$

Step-by-step solution

Expand the cross product

We need to expand the cross product \((\mathbf{F} \times \mathbf{G})\), which can be written as: $$(\mathbf{F} \times \mathbf{G})= \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ F_{x} & F_{y} & F_{z} \\ G_{x} & G_{y} & G_{z} \end{vmatrix}$$ Now, find the components of the cross product: $$\mathbf{F} \times \mathbf{G} = (F_{y}G_{z} - F_{z}G_{y})\mathbf{i} - (F_{x}G_{z} - F_{z}G_{x})\mathbf{j} + (F_{x}G_{y} - F_{y}G_{x})\mathbf{k}$$

Compute the divergence

Now compute the divergence of the cross product, which can be written as: $$\nabla \cdot (\mathbf{F} \times \mathbf{G}) = \frac{\partial}{\partial x}\left(F_{y}G_{z} - F_{z}G_{y}\right) + \frac{\partial}{\partial y}\left(G_{x}F_{z} - F_{x}G_{z}\right) + \frac{\partial}{\partial z}\left(F_{x}G_{y} - F_{y}G_{x}\right)$$

Compute the curls

Compute the curls of both vector fields, which can be written as: $$\begin{aligned} \nabla \times \mathbf{F} &= \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ F_{x} & F_{y} & F_{z} \end{vmatrix}, \\ \nabla \times \mathbf{G} &= \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ G_{x} & G_{y} & G_{z} \end{vmatrix} \end{aligned}$$ Now find the components of the curls: $$\begin{aligned} \nabla \times \mathbf{F} &= \left(\frac{\partial F_{z}}{\partial y} - \frac{\partial F_{y}}{\partial z}\right)\mathbf{i} - \left(\frac{\partial F_{z}}{\partial x} - \frac{\partial F_{x}}{\partial z}\right)\mathbf{j} + \left(\frac{\partial F_{y}}{\partial x} - \frac{\partial F_{x}}{\partial y}\right)\mathbf{k}, \\ \nabla \times \mathbf{G} &= \left(\frac{\partial G_{z}}{\partial y} - \frac{\partial G_{y}}{\partial z}\right)\mathbf{i} - \left(\frac{\partial G_{z}}{\partial x} - \frac{\partial G_{x}}{\partial z}\right)\mathbf{j} + \left(\frac{\partial G_{y}}{\partial x} - \frac{\partial G_{x}}{\partial y}\right)\mathbf{k} \end{aligned}$$

Compute the scalar product

Compute the scalar products of the vector fields with the curls: $$\begin{aligned} \mathbf{G} \cdot (\nabla \times \mathbf{F}) &= G_{x}\left(\frac{\partial F_{z}}{\partial y} - \frac{\partial F_{y}}{\partial z}\right) - G_{y}\left(\frac{\partial F_{z}}{\partial x} - \frac{\partial F_{x}}{\partial z}\right) + G_{z}\left(\frac{\partial F_{y}}{\partial x} - \frac{\partial F_{x}}{\partial y}\right), \\ \mathbf{F} \cdot (\nabla \times \mathbf{G}) &= F_{x}\left(\frac{\partial G_{z}}{\partial y} - \frac{\partial G_{y}}{\partial z}\right) - F_{y}\left(\frac{\partial G_{z}}{\partial x} - \frac{\partial G_{x}}{\partial z}\right) + F_{z}\left(\frac{\partial G_{y}}{\partial x} - \frac{\partial G_{x}}{\partial y}\right) \end{aligned}$$

Simplify the identity

Finally, compute the difference of the scalar products: $$\mathbf{G} \cdot(\nabla \times \mathbf{F})-\mathbf{F} \cdot(\nabla \times \mathbf{G}) = \frac{\partial}{\partial x}\left(F_{y}G_{z} - F_{z}G_{y}\right) + \frac{\partial}{\partial y}\left(G_{x}F_{z} - F_{x}G_{z}\right) + \frac{\partial}{\partial z}\left(F_{x}G_{y} - F_{y}G_{x}\right)$$ From steps 2 and 5, we find that both expressions are the same, thus proving the identity: $$\nabla \cdot(\mathbf{F} \times \mathbf{G})=\mathbf{G} \cdot(\nabla \times \mathbf{F})-\mathbf{F} \cdot(\nabla \times \mathbf{G})$$

Key concepts

Cross Product

The cross product is a fundamental operation in vector calculus. It's used to find a vector that is perpendicular to two given vectors in three-dimensional space. This operation is particularly useful in physics and engineering. When calculating the cross product of two vectors \( \mathbf{F} \) and \( \mathbf{G} \), the resulting vector is orthogonal to both \( \mathbf{F} \) and \( \mathbf{G} \).
This vector is represented as the determinant of a 3x3 matrix formed by the unit vectors \( \mathbf{i}, \mathbf{j}, \mathbf{k} \) and the components of \( \mathbf{F} \) and \( \mathbf{G} \):

• \((\mathbf{F} \times \mathbf{G}) = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ F_x & F_y & F_z \ G_x & G_y & G_z \end{vmatrix} \)

By solving the determinant, you can find each of the components:

• In the \( \mathbf{i} \) direction: \( F_yG_z - F_zG_y \)
• In the \( \mathbf{j} \) direction: \( -(F_xG_z - F_zG_x) \)
• In the \( \mathbf{k} \) direction: \( F_xG_y - F_yG_x \)

Understanding the cross product is crucial as it's used to calculate other vector operations like curl.

Vector Fields

Vector fields assign a vector to every point in a region of space. These vectors can represent anything from wind speeds and directions to magnetic fields. Understanding vector fields is critical because they describe how vectors behave in a particular region.
In mathematics, vector fields are often represented by functions, for instance, \( \mathbf{F}(x, y, z) = (F_x, F_y, F_z) \). These functions define the vector at any given point \( (x, y, z) \) in three-dimensional space.
Vector fields are key in various scientific fields:

• Physics: To describe fields like gravitational or electromagnetic fields.
• Fluid dynamics: To model the flow of air or liquid around objects.
• Engineering: For stress analyses and other applications involving forces at multiple points.

Analyzing vector fields often involves looking at other properties like divergence and curl, which describe how the vector field behaves in space.

Divergence

Divergence is a scalar value that measures the magnitude of a source or sink at a given point in a vector field. In simpler terms, it tells us how much "stuff" is expanding out of or converging into a point.
The divergence of a vector field \( \mathbf{F} = (F_x, F_y, F_z) \) is computed as:\[ abla \cdot \mathbf{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z} \]

• A positive divergence indicates a source, where the vector field disperses outward.
• A negative divergence indicates a sink, where vectors converge inward.

Divergence is a critical concept in the identity discussed in this exercise, where we find the divergence of the cross product. Understanding it helps explain phenomena in fields like fluid dynamics and electromagnetic theory.

Curl

The curl of a vector field provides a measure of the rotation or the "twist" of the field around a point. It is also a vector, offering insight into the rotational characteristics of the vector field.
To compute the curl of a vector field \( \mathbf{F} = (F_x, F_y, F_z) \), use:\[ abla \times \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \ F_x & F_y & F_z \end{vmatrix} \] The components of this determinant represent the rotational action of the vector field:

• In \( \mathbf{i} \): \( \frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z} \)
• In \( \mathbf{j} \): \( \frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x} \)
• In \( \mathbf{k} \): \( \frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y} \)

Curl is essential in fields such as electromagnetism, where it helps describe how magnetic and electric fields interact. It is also crucial in fluid dynamics to model vortices and rotational flow.