Question

Show that if \(f\) and \(g\) are scalars having continuous second partial derivatives in a domain \(D\), then $$ \mathbf{u}=\boldsymbol{\nabla} f \times \nabla g $$ is solenoidal in \(D\). (It can be shown that every solenoidal vector has such a reprefentation, at least in a suitably restricted domain.)

Short answer

Question: Show that the vector field \(\mathbf{u}\) given by \(\mathbf{u}=\nabla f \times \nabla g\) is solenoidal in the domain \(D\). Answer: The divergence of \(\mathbf{u}\) is zero in \(D\), so \(\mathbf{u}\) is solenoidal.

Step-by-step solution

Calculate the cross product of the gradients

First, let's calculate the vector field \(\mathbf{u}\), which is defined as the cross product of the gradients of \(f\) and \(g\). $$ \mathbf{u} = \nabla f \times \nabla g $$

Calculate the curl of the gradients

In order to use the first identity mentioned above, we need to calculate the curl of the gradients of \(f\) and \(g\). $$ \nabla \times (\nabla f) = \mathbf{0} $$ $$ \nabla \times (\nabla g) = \mathbf{0} $$ Note that the curl of the gradient of any scalar field is always zero.

Apply the identity

Now that we know the curl of the gradients, we can use the identity mentioned above to calculate the divergence of the cross product of the gradients. $$ \nabla \cdot \mathbf{u} = \nabla \cdot (\nabla f \times \nabla g) = \nabla g \cdot (\nabla \times \nabla f) - \nabla f \cdot (\nabla \times \nabla g) $$

Plug in the curls and simplify

We now substitute the curls we calculated earlier and simplify the expression. $$ \nabla \cdot \mathbf{u} = \nabla g \cdot \mathbf{0} - \nabla f \cdot \mathbf{0} = 0 $$

Conclusion

Since the divergence of \(\mathbf{u}\) is zero for all points in the domain \(D\), it is a solenoidal vector field. This confirms that $$ \mathbf{u} = \nabla f \times \nabla g $$ is indeed solenoidal in \(D\).

Key concepts

Cross Product of Gradients

In vector calculus, the cross product of gradients is a powerful tool used to create a vector field from two scalar functions. To compute the cross product of the gradients of two scalar functions, say \( f \) and \( g \), we first need to calculate their respective gradients, \( abla f \) and \( abla g \).

The gradient of a scalar function is essentially a vector that points in the direction of the steepest increase of the function. For a function \( f(x, y, z) \), the gradient is calculated as \( abla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right) \).
Similarly, \( abla g \) is the gradient of the function \( g(x, y, z) \).

• To find the cross product \( \mathbf{u} = abla f \times abla g \), use the determinant of a matrix constructed from the unit vectors \( \mathbf{i}, \mathbf{j}, \mathbf{k} \), the components of \( abla f \), and \( abla g \).
• This cross product \( \mathbf{u} \) is a vector field that is orthogonal to both \( abla f \) and \( abla g \).

Continuous Second Partial Derivatives

For scalar functions \( f \) and \( g \) to consistently define a vector field through their gradients, they are required to have continuous second partial derivatives in their domain \( D \).

Continuous second partial derivatives mean that all second-order derivatives of the functions are continuous in that domain. Mathematically, this means:

• \( \frac{\partial^2 f}{\partial x^2} \), \( \frac{\partial^2 f}{\partial x\partial y} \), \( \frac{\partial^2 f}{\partial x\partial z} \), and so on, are continuous for \( f \).
• Similarly continuous derivatives exist for \( g \).

The existence of continuous partial derivatives ensures that these functions are smooth, that helps avoid undefined gradient values at any point in the domain.
This property is crucial when applying theorems like Green's theorem or the Divergence theorem, which rely on smoothness for calculus operations.

Divergence

Divergence is an operator that measures the extent to which a vector field acts as a source or a sink at a given point.
For a vector field \( \mathbf{v} = (v_1, v_2, v_3) \), the divergence is calculated as \( abla \cdot \mathbf{v} = \frac{\partial v_1}{\partial x} + \frac{\partial v_2}{\partial y} + \frac{\partial v_3}{\partial z} \).

In context, for \( \mathbf{u} = abla f \times abla g \), we want to determine if \( \mathbf{u} \) is solenoidal, meaning it has zero divergence everywhere in \( D \).
With:

• The property that the curl of any gradient is zero: \( abla \times abla f = \mathbf{0} \)
• The identity \( abla \cdot (abla f \times abla g) = abla g \cdot (abla \times abla f) - abla f \cdot (abla \times abla g) \)

When you apply these, the divergence simplifies to zero, confirming the solenoidal nature of \( \mathbf{u} \).
Thus, \( \mathbf{u} \), having zero divergence everywhere in \( D \), does not act as a source or a sink.

Curl of a Gradient

The curl of a gradient is always a noteworthy concept in vector calculus because it simplifies many problems.
When we calculate the curl of the gradient of a scalar function \( f \), we get:

• \( abla \times (abla f) = \mathbf{0} \)

This result holds because differentiation is commutative; hence the mixed second partial derivatives cancel each other out.

This property is instrumental in simplifying the problem of determining if a vector field like \( \mathbf{u} \) is solenoidal. Since \( \mathbf{u} = abla f \times abla g \) involves the curl of \( abla f \) and \( abla g \), both being zero, it leads directly to the conclusion that \( \mathbf{u} \) is solenoidal.
Understanding this concept saves a lot of time and computation in both theoretical and practical applications of vector calculus.