Problem 21 - Advanced Engineering Mathematics

Question

Let \(\mathbf{a}\) be a constant voctor and \(\mathbf{r}=x \mathbf{i}+y \mathbf{j}+z \mathbf{k}\). Verify the given identity. $$ \nabla \cdot(\mathbf{a} \times \mathbf{r})=0 $$

Step-by-Step Solution

Answer

The divergence of \( \mathbf{a} \times \mathbf{r} \) is zero, confirming the identity.

Step 1: Understanding the Given Expression

We need to verify that the divergence of the cross product of a constant vector \( \mathbf{a} \) and the vector \( \mathbf{r} \) is zero. The expression is \( abla \cdot(\mathbf{a} \times \mathbf{r})=0 \), where \( abla \) is the del operator, \( \mathbf{a} = a_x \mathbf{i} + a_y \mathbf{j} + a_z \mathbf{k} \) is a constant vector, and \( \mathbf{r} = x \mathbf{i} + y \mathbf{j} + z \mathbf{k} \).

Step 2: Calculating the Cross Product

First, compute the cross product \( \mathbf{a} \times \mathbf{r} \). Using the determinant method:\[\mathbf{a} \times \mathbf{r} = \begin{vmatrix}\mathbf{i} & \mathbf{j} & \mathbf{k} \a_x & a_y & a_z \x & y & z \\end{vmatrix}\]Which gives:\[= (a_y z - a_z y) \mathbf{i} - (a_x z - a_z x) \mathbf{j} + (a_x y - a_y x) \mathbf{k}\]Thus, \( \mathbf{a} \times \mathbf{r} = (a_y z - a_z y) \mathbf{i} - (a_x z - a_z x) \mathbf{j} + (a_x y - a_y x) \mathbf{k} \).

Step 3: Compute the Divergence of the Cross Product

Now, apply the del operator to find the divergence of \( \mathbf{a} \times \mathbf{r} \), which is achieved by finding the dot product \( abla \cdot (\mathbf{a} \times \mathbf{r}) \):\[abla \cdot (\mathbf{a} \times \mathbf{r}) = \frac{\partial}{\partial x}(a_y z - a_z y) + \frac{\partial}{\partial y}(-(a_x z - a_z x)) + \frac{\partial}{\partial z}(a_x y - a_y x)\]Compute each partial derivative:- \( \frac{\partial}{\partial x}(a_y z - a_z y) = 0 \) since \( a_y z \) and \( a_z y \) do not depend on \( x \).- \( \frac{\partial}{\partial y}(-(a_x z - a_z x)) = 0 \) since \( a_x z \) and \( a_z x \) do not depend on \( y \).- \( \frac{\partial}{\partial z}(a_x y - a_y x) = 0 \) since \( a_x y \) and \( a_y x \) do not depend on \( z \).

Step 4: Conclusion

Since all partial derivatives are zero, we verify that:\[abla \cdot (\mathbf{a} \times \mathbf{r}) = 0\]Thus, the identity is verified, showing that the divergence of the cross product of \( \mathbf{a} \) and \( \mathbf{r} \) is indeed zero.

Key Concepts

Gradient

In vector calculus, the gradient is a vector operator that operates on a scalar field to produce a vector field. It is denoted by the symbol \( abla \) and is called "del." Imagine the gradient as a way to determine the slope of a hill you're climbing if you are modeling the hill using a height function.

• The gradient points in the direction of the greatest increase of a function.
• The magnitude of the gradient vector indicates the steepness of the incline.

For a function \( f(x, y, z) \), the gradient is expressed as: \[ abla f = \left( \frac{\partial f}{\partial x} \right)\mathbf{i} + \left( \frac{\partial f}{\partial y} \right)\mathbf{j} + \left( \frac{\partial f}{\partial z} \right)\mathbf{k} \]The gradient is fundamental in optimization and helps find critical points where functions have minima, maxima, or saddle points. It also holds a deep connection to lines of force in physics, with applications in gravitation, electromagnetics, and flow dynamics.

Divergence

Divergence is another vector operator used in vector calculus, often applied to vector fields. It measures the magnitude of a field's source or sink at a given point, effectively quantifying how much a vector field "diverges" or "spreads out" from that point. Think of divergence as how fluid flows out of a point.

• If the divergence at a point is positive, more of the field is flowing out than flowing in.
• If the divergence is negative, more of the field is converging into the point.
• A zero divergence means the field is neither expanding nor contracting at that point.

For a vector field \( \mathbf{F} = P \mathbf{i} + Q \mathbf{j} + R \mathbf{k} \), divergence is given by:\[ abla \cdot \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z} \]Divergence has several applications, chiefly in electromagnetic theory, fluid dynamics, and heat transfer. When applied to the cross product of vectors, as seen in the problem, divergence can lead to fascinating results, such as showing that certain vector changes sum to zero.

Cross Product

The cross product is a binary operation on two vectors in three-dimensional space, resulting in another vector that is orthogonal to both original vectors. It's key in various applications, especially in physics and 3D graphics, as it provides insight into rotational effects and angular momentum.

• The cross product produces a vector whose direction is determined by the right-hand rule.
• The magnitude of the cross product vector is equal to the area of the parallelogram formed by the two vectors.

For vectors \( \mathbf{a} = a_x \mathbf{i} + a_y \mathbf{j} + a_z \mathbf{k} \) and \( \mathbf{b} = b_x \mathbf{i} + b_y \mathbf{j} + b_z \mathbf{k} \), the cross product \( \mathbf{a} \times \mathbf{b} \) is calculated by:\[\mathbf{a} \times \mathbf{b} = (a_y b_z - a_z b_y) \mathbf{i} - (a_x b_z - a_z b_x) \mathbf{j} + (a_x b_y - a_y b_x) \mathbf{k}\]In the provided exercise, calculating the cross product and taking its divergence reveals an intriguing property of vector fields: the underlying structures sometimes result in zero divergence, signaling no net "outflux" from a spatial point. This principle is pivotal in various theoretical and practical physics contexts.