Problem 20 - 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. $$ \nabla \times(\mathbf{a} \times \mathbf{r})=2 \mathbf{a} $$

Step-by-Step Solution

Answer

Yes, the identity is verified: \( \nabla \times (\mathbf{a} \times \mathbf{r}) = 2 \mathbf{a} \).

Step 1: Understand the Cross Product

The given identity involves cross products. The cross product \( \mathbf{a} \times \mathbf{r} \) is a vector that is perpendicular to both \( \mathbf{a} \) and \( \mathbf{r} \). In this specific case, \( \mathbf{r} = x \mathbf{i} + y \mathbf{j} + z \mathbf{k} \). We need to first compute \( \mathbf{a} \times \mathbf{r} \) using the determinant method.

Step 2: Compute \( \mathbf{a} \times \mathbf{r} \)

Assume \( \mathbf{a} = a_x \mathbf{i} + a_y \mathbf{j} + a_z \mathbf{k} \). The cross product \( \mathbf{a} \times \mathbf{r} \) is expressed as the determinant:\[\mathbf{a} \times \mathbf{r} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ a_x & a_y & a_z \ x & y & z \end{vmatrix} = (a_y z - a_z y) \mathbf{i} - (a_x z - a_z x) \mathbf{j} + (a_x y - a_y x) \mathbf{k}.\]

Step 3: Apply the Curl to \( \mathbf{a} \times \mathbf{r} \)

The curl of a vector \( \mathbf{F} = F_x \mathbf{i} + F_y \mathbf{j} + F_z \mathbf{k} \) is given by:\[abla \times \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \ F_x & F_y & F_z \end{vmatrix}.\]For \( \mathbf{a} \times \mathbf{r} \), we have:\[ F_x = a_y z - a_z y, \quad F_y = -(a_x z - a_z x), \quad F_z = a_x y - a_y x. \]

Step 4: Compute \( \nabla \times (\mathbf{a} \times \mathbf{r}) \)

Compute each component of the curl:- For \( \mathbf{i} \): \( \frac{\partial}{\partial y}(a_x y - a_y x) - \frac{\partial}{\partial z}(-(a_x z - a_z x)) = a_x - (a_x) = 0 \).- For \( \mathbf{j} \): \( \frac{\partial}{\partial z}(a_y z - a_z y) - \frac{\partial}{\partial x}(a_x y - a_y x) = a_y - (-a_y) = 2a_y \).- For \( \mathbf{k} \): \( \frac{\partial}{\partial x}(-(a_x z - a_z x)) - \frac{\partial}{\partial y}(a_y z - a_z y) = a_z - (-a_z) = 2a_z \).Thus, \[ abla \times (\mathbf{a} \times \mathbf{r}) = 0 \mathbf{i} + 2a_y \mathbf{j} + 2a_z \mathbf{k} = 2 \mathbf{a}. \]

Step 5: Verify the Result

The computed curl \( abla \times (\mathbf{a} \times \mathbf{r}) \) results in \( 2 \mathbf{a} \), which confirms the given identity. Therefore, the verification is complete.

Key Concepts

Cross Product

The cross product is a vector operation used in three-dimensional space. It is denoted by the symbol \( \times \) between two vectors. The result of a cross product is a new vector that is perpendicular to the plane containing the original pair of vectors.
To perform a cross product between two vectors, such as a constant vector \( \mathbf{a} = a_x \mathbf{i} + a_y \mathbf{j} + a_z \mathbf{k} \) and the position vector \( \mathbf{r} = x \mathbf{i} + y \mathbf{j} + z \mathbf{k} \), we use the formula of the determinant:
\[\mathbf{a} \times \mathbf{r} = \begin{vmatrix}\mathbf{i} & \mathbf{j} & \mathbf{k} \ a_x & a_y & a_z \ x & y & z\end{vmatrix} = (a_y z - a_z y) \mathbf{i} - (a_x z - a_z x) \mathbf{j} + (a_x y - a_y x) \mathbf{k}.\]

• Determinant method simplifies the calculations.
• Cross product gives direction perpendicular to both vectors.

Understanding the cross product is crucial for vector calculus problems involving perpendicularity and vector fields.

Curl of a Vector

In the realm of vector calculus, the curl of a vector is a significant operator. It measures the rotational motion or circulation of a vector field at a point. For a vector field \( \mathbf{F} = F_x \mathbf{i} + F_y \mathbf{j} + F_z \mathbf{k} \), the curl is given by:
\[abla \times \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \F_x & F_y & F_z \end{vmatrix}.\]

• It involves the partial derivatives of the components of the vector field.
• The result is also a vector indicating local rotation.

Applying this concept to the vector obtained by the cross product, \( \mathbf{a} \times \mathbf{r} \), one computes the rotational tendency of this vector field. This helps in understanding phenomena like magnetic fields and fluid dynamics.

Vector Identity Verification

Verifying vector identities is an essential practice in vector calculus. It involves proving that one side of an equation transforms into the other through established vector operations or identities.
To verify the identity \( abla \times (\mathbf{a} \times \mathbf{r}) = 2 \mathbf{a} \), follow these steps:
1. Compute the cross product \( \mathbf{a} \times \mathbf{r} \) using determinants.2. Apply the curl operation to \( \mathbf{a} \times \mathbf{r} \) by computing its partial derivatives.3. Check if the resulting vector matches \( 2 \mathbf{a} \).
Through this systematic verification of vector identities, we confirm our understanding of the relationships between different vector operations. This is crucial for fields like electromagnetism and fluid mechanics, where such identities simplify complex relationships.