Problem 31 - Thomas Calculus

Question

If \(\mathbf{F}=M \mathbf{i}+N \mathbf{j}+P \mathbf{k}\) is a differentiable vector field, we define the notation \(\mathbf{F} \cdot \nabla\) to mean $$M \frac{\partial}{\partial x}+N \frac{\partial}{\partial y}+P \frac{\partial}{\partial z}$$ For differentiable vector fields \(\mathbf{F}_{1}\) and \(\mathbf{F}_{2},\) verify the following identities. $$\begin{array}{l}{\text { a. } \nabla \times\left(\mathbf{F}_{1} \times \mathbf{F}_{2}\right)=\left(\mathbf{F}_{2} \cdot \nabla\right) \mathbf{F}_{1}-\left(\mathbf{F}_{1} \cdot \nabla\right) \mathbf{F}_{2}+\left(\nabla \cdot \mathbf{F}_{2}\right) \mathbf{F}_{1}-} \\\ {\quad\left(\nabla \cdot \mathbf{F}_{1}\right) \mathbf{F}_{2}} \\ {\mathbf{b} . \nabla\left(\mathbf{F}_{1} \cdot \mathbf{F}_{2}\right)=\left(\mathbf{F}_{1} \cdot \nabla\right) \mathbf{F}_{2}+\left(\mathbf{F}_{2} \cdot \nabla\right) \mathbf{F}_{1}+\mathbf{F}_{1} \times\left(\nabla \times \mathbf{F}_{2}\right)+} \\ {\mathbf{F}_{2} \times\left(\nabla \times \mathbf{F}_{1}\right)}\end{array}$$

Step-by-Step Solution

Answer

Both identities follow directly from established vector calculus identities.

Step 1: Understand the problem

We are given two differentiable vector fields, \( \mathbf{F}_1 \) and \( \mathbf{F}_2 \), and asked to verify two vector calculus identities involving operations such as divergence, gradient, and curl. The identities involve complex vector operations which require a systematic verification process.

Step 2: Use Vector Identity Properties

Before solving, understand the key vector calculus operations: Divergence \( abla \cdot \), Curl \( abla \times \), and Gradient \( abla f \). Each property has its formulaic representation which will be needed in manipulations.

Step 3: Apply the Double Cross Product Identity for Part (a)

Use the vector identity \( abla \times (\mathbf{A} \times \mathbf{B}) = (\mathbf{B} \cdot abla) \mathbf{A} - (\mathbf{A} \cdot abla) \mathbf{B} + (abla \cdot \mathbf{B}) \mathbf{A} - (abla \cdot \mathbf{A}) \mathbf{B} \). This directly aligns with the given identity in part (a).

Step 4: Verify Part (a)

Check each term from the identity \( abla \times (\mathbf{F}_1 \times \mathbf{F}_2) = (\mathbf{F}_2 \cdot abla) \mathbf{F}_1 - (\mathbf{F}_1 \cdot abla) \mathbf{F}_2 + (abla \cdot \mathbf{F}_2) \mathbf{F}_1 - (abla \cdot \mathbf{F}_1) \mathbf{F}_2 \). Verify that performing each operation yields the result according to the rules of vector calculus.

Step 5: Apply Gradient of a Dot Product Identity for Part (b)

Use the identity \( abla (\mathbf{A} \cdot \mathbf{B}) = (\mathbf{A} \cdot abla) \mathbf{B} + (\mathbf{B} \cdot abla) \mathbf{A} + \mathbf{A} \times (abla \times \mathbf{B}) + \mathbf{B} \times (abla \times \mathbf{A}) \). This corresponds to the algebraic structure of part (b).

Step 6: Verify Part (b)

Perform derivatives and cross products for terms according to the identity: \( abla (\mathbf{F}_1 \cdot \mathbf{F}_2) = (\mathbf{F}_1 \cdot abla) \mathbf{F}_2 + (\mathbf{F}_2 \cdot abla) \mathbf{F}_1 + \mathbf{F}_1 \times (abla \times \mathbf{F}_2) + \mathbf{F}_2 \times (abla \times \mathbf{F}_1) \). Compute each term individually to confirm.

Key Concepts

Differentiable Vector Fields

In vector calculus, a differentiable vector field is a function that assigns a vector to every point in a region of space and has smooth (continuous) derivatives. Vector fields are paramount in expressing physical quantities such as velocity fields, electromagnetic fields, and gravitational fields. These fields can be represented by functions like \(\mathbf{F} = M \mathbf{i} + N \mathbf{j} + P \mathbf{k}\), where \(M, N,\) and \(P\) are differentiable functions of spatial coordinates \(x, y,\) and \(z\).
This essentially means that the derivatives \(\frac{\partial M}{\partial x}, \frac{\partial N}{\partial y}, \) and \(\frac{\partial P}{\partial z}\) are well-defined and continuous over the region in question. This property of differentiability is crucial when applying operations such as divergence, curl, and gradient on vector fields, as it ensures the meaningfulness and continuity of the outcomes during mathematical manipulation.
For example, in verifying vector calculus identities like those given in the exercise, the differentiability condition is relied upon to apply any derivative and integrate throughout the region.

Gradient

The gradient is a vector operation that measures the rate and direction of change in a scalar field (usually a function) at any point. It generates a vector from a scalar input by combining partial derivatives, thus becoming an essential tool in scalar field analysis.
The gradient operator is denoted by \(abla\), and for a scalar function \(f(x, y, z)\), it is expressed as:
\[ abla f = \frac{\partial f}{\partial x} \mathbf{i} + \frac{\partial f}{\partial y} \mathbf{j} + \frac{\partial f}{\partial z} \mathbf{k} \]

• The gradient points in the direction of the most rapid increase of the function.
• Its magnitude is the rate of maximum increase per unit distance.

In vector calculus, understanding the gradient allows you to determine how a scalar field, like temperature or pressure, changes spatially. When applied in vector identities, the gradient helps in breaking down complex interactions between fields, such as in part b of the exercise, enriching our understanding of the relationships between scalar fields and vectors.

Curl

Curl is a vector operator that describes the rotation of a vector field inside a region of space. It provides insight into the swirling and rotational behavior of the field at any point, especially in fluid dynamics and electromagnetism.
The curl of a vector field \(\mathbf{F} = M \mathbf{i} + N \mathbf{j} + P \mathbf{k}\) is given by:
\[ abla \times \mathbf{F} = \left( \frac{\partial P}{\partial y} - \frac{\partial N}{\partial z} \right) \mathbf{i} + \left( \frac{\partial M}{\partial z} - \frac{\partial P}{\partial x} \right) \mathbf{j} + \left( \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} \right) \mathbf{k} \]

• When the curl is zero, the field is irrotational.
• The direction of the curl vector indicates the axis of rotation, and its magnitude indicates the speed of rotation.

The concept of curl plays a critical role in dynamics and is used extensively in problems involving the circulation of fluids or currents. In part a of the exercise, understanding and applying the curl help verify and uncover interactions between intertwined vectors, keeping the analysis grounded despite the complexity.

Divergence

Divergence is a scalar operation that measures the magnitude of a vector field's source or sink at a given point. It reflects how much a vector field spreads out or diverges from any point in its domain.
The divergence of a vector field \(\mathbf{F} = M \mathbf{i} + N \mathbf{j} + P \mathbf{k}\) is given by:

• \(abla \cdot \mathbf{F} = \frac{\partial M}{\partial x} + \frac{\partial N}{\partial y} + \frac{\partial P}{\partial z}\)

Its value provides vital information about the field behavior:

• If the divergence is positive, the field behaves like a source, emanating vectors outward.
• A negative divergence indicates a sink, drawing vectors inward.
• Zero divergence implies the vector field is solenoidal, indicating no net flow across a surface.

In calculus exercises, divergence helps in simplifying complex expressions, especially when used alongside other vector operations. In the problem's context, as seen in exploring the identities, divergence validates the expansion and contraction dynamics present in multidimensional vector fields.