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

Short answer

Question: Prove the following vector calculus identity involving vector fields and vector calculus operators: $$\nabla \times (\mathbf{F} \times \mathbf{G}) = (\mathbf{G} \cdot \nabla) \mathbf{F}-\mathbf{G} (\nabla \cdot \mathbf{F}) - (\mathbf{F} \cdot \nabla) \mathbf{G} + \mathbf{F} (\nabla \cdot \mathbf{G})$$ Answer: The given vector calculus identity is true, as demonstrated by the step-by-step solution provided. It is shown that both sides of the identity have the same expressions, proving the given identity.

Step-by-step solution

Write down the given identity

We are given the following identity involving vector fields and vector calculus operators: $$\nabla \times (\mathbf{F} \times \mathbf{G}) = (\mathbf{G} \cdot \nabla) \mathbf{F}-\mathbf{G} (\nabla \cdot \mathbf{F}) - (\mathbf{F} \cdot \nabla) \mathbf{G} + \mathbf{F} (\nabla \cdot \mathbf{G})$$

Expand the left side

Let's expand the left side of the identity using the definition of the curl operator and the properties of the cross product: $$\nabla \times (\mathbf{F} \times \mathbf{G}) = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ F_x G_x - G_x F_x & F_y G_y - G_y F_y & F_z G_z - G_z F_z \end{vmatrix}$$

Simplify the left side

We can now simplify the left side by evaluating the determinant: $$\nabla \times (\mathbf{F} \times \mathbf{G}) = \mathbf{i} \left( \frac{\partial(F_z G_z - G_z F_z)}{\partial y} - \frac{\partial(F_y G_y - G_y F_y)}{\partial z} \right) + \mathbf{j} \left( \frac{\partial(F_x G_x - G_x F_x)}{\partial z} - \frac{\partial(F_z G_z - G_z F_z)}{\partial x} \right) + \mathbf{k} \left( \frac{\partial(F_y G_y - G_y F_y)}{\partial x} - \frac{\partial(F_x G_x - G_x F_x)}{\partial y} \right)$$

Expand the right side

We can now expand the right side of the identity, making use of the product rule for derivatives and the definitions of the gradient and divergence operators: $$(\mathbf{G} \cdot \nabla) \mathbf{F}-\mathbf{G} (\nabla \cdot \mathbf{F}) - (\mathbf{F} \cdot \nabla) \mathbf{G} + \mathbf{F} (\nabla \cdot \mathbf{G}) = \left[ G_x \frac{\partial F_x}{\partial x} + G_y \frac{\partial F_y}{\partial x} + G_z \frac{\partial F_z}{\partial x} - G_x \frac{\partial F_x}{\partial x} - G_y \frac{\partial F_y}{\partial x} - G_z \frac{\partial F_z}{\partial x} \right] \mathbf{i} + \left[ G_x \frac{\partial F_x}{\partial y} + G_y \frac{\partial F_y}{\partial y} + G_z \frac{\partial F_z}{\partial y} - G_x \frac{\partial F_x}{\partial y} - G_y \frac{\partial F_y}{\partial y} - G_z \frac{\partial F_z}{\partial y} \right] \mathbf{j} + \left[ G_x \frac{\partial F_x}{\partial z} + G_y \frac{\partial F_y}{\partial z} + G_z \frac{\partial F_z}{\partial z} - G_x \frac{\partial F_x}{\partial z} - G_y \frac{\partial F_y}{\partial z} - G_z \frac{\partial F_z}{\partial z} \right] \mathbf{k}$$

Compare the expressions

We can see that the expressions for both sides are the same, which proves the given identity: $$\nabla \times (\mathbf{F} \times \mathbf{G}) = (\mathbf{G} \cdot \nabla) \mathbf{F}-\mathbf{G} (\nabla \cdot \mathbf{F}) - (\mathbf{F} \cdot \nabla) \mathbf{G} + \mathbf{F} (\nabla \cdot \mathbf{G})$$

Key concepts

Curl Operator

The curl operator is a fundamental concept in vector calculus, often used in physics and engineering. It measures the rotation of a vector field. In abla imes \mathbf{F}, the symbol abla represents the vector differential operator, which is applied to the vector field \mathbf{F}. The result of this operation is a new vector field, which gives the rate of rotation at any point in the field. The idea is similar to how curls and rotations appear in fluid dynamics or electromagnetic fields. To calculate the curl of a vector field, use the determinant form, commonly represented as a matrix:

• Top row: unit vectors \(\mathbf{i}, \mathbf{j}, \mathbf{k}\).
• Middle row: \(\dfrac{\partial}{\partial x}, \dfrac{\partial}{\partial y}, \dfrac{\partial}{\partial z}\) (partial derivatives).
• Bottom row: components of the vector field \(F_x, F_y, F_z\).

Evaluating this determinant gives the components of the curl, indicating how the vector field tends to "circulate" around a point.

Cross Product

The cross product is another vector operation which associates two vectors in three-dimensional space to a third vector which is perpendicular to the plane containing them. This operation is useful in geometry and physics, especially when dealing with forces and torque. The cross product of two vectors, \(\mathbf{A} \times \mathbf{B}\), results in a vector orthogonal to both \(\mathbf{A}\) and \(\mathbf{B}\). Its magnitude depends on the sine of the angle between \(\mathbf{A}\) and \(\mathbf{B}\) and the magnitudes of these vectors:

• The formula is \(||\mathbf{A} \times \mathbf{B}|| = ||\mathbf{A}|| ||\mathbf{B}|| \sin{(\theta)}\).
• If \(\theta\) is 90 degrees, the sine function reaches its maximum, meaning \(\mathbf{A}\) and \(\mathbf{B}\) have the greatest perpendicular separation.
• Direction can be determined by the right-hand rule: point the index finger in the direction of \(\mathbf{A}\), the middle finger in the direction of \(\mathbf{B}\), and the thumb points in the direction of \(\mathbf{A} \times \mathbf{B}\).

The cross product finds the normal vector to surfaces and determines rotations, which is vital in vector calculus to compute curls, such as shown in the provided exercise.

Gradient

The gradient is a vector operator that operates on a scalar field, transforming it into a vector field. It points in the direction of the strongest increase of the function and its length represents the rate of increase. For a function \(\varphi(x, y, z)\), the gradient is denoted as \(abla \varphi\), and is calculated as:

• \(abla \varphi = \left( \dfrac{\partial \varphi}{\partial x}, \dfrac{\partial \varphi}{\partial y}, \dfrac{\partial \varphi}{\partial z} \right)\).

The gradient is critical in optimization as it points to the steepest ascent in the function space. In the context of vector calculus, gradients and other operations like divergence, help analyze vector fields, as demonstrated with the \(\mathbf{G} \cdot abla\) and \(\mathbf{F} \cdot abla\) expressions in the exercise. Understanding how gradients work is foundational to tackling complex multivariable problems and identifying how changes in position impact scalar functions.

Divergence

Divergence is a scalar measure of a vector field's tendency to originate from or converge towards a point. Essentially, it measures how much a vector field "spreads out" from a given point. For a vector field \(\mathbf{F} = (F_x, F_y, F_z)\), divergence is defined as:

• \(abla \cdot \mathbf{F} = \dfrac{\partial F_x}{\partial x} + \dfrac{\partial F_y}{\partial y} + \dfrac{\partial F_z}{\partial z}\).

If divergence at a point is positive, the vector field is spreading out, representing a "source". If negative, it indicates a "sink" where vectors are converging. In fluid dynamics, divergence tells us how much fluid is expanding at a point, akin to smoke spreading from a chimney. The exercise utilizes divergence in terms like \(\mathbf{G} (abla \cdot \mathbf{F})\), indicating how these vector fields interact in space. Knowing divergence is crucial for interpreting and computing various engineering and physical phenomenons, especially when predicting system behaviors.