Problem 41 - Calculus Early Transcendentals

Question

Verify that the radius vector r = xi + y j + zk has the stated property. $$ \text { (a) curl } \mathbf{r}=\mathbf{0} \quad \text { (b) } \nabla\|\mathbf{r}\|=\frac{\mathbf{r}^{*}}{\|\mathbf{r}\|} $$

Step-by-Step Solution

Answer

(a) \( \nabla \times \mathbf{r} = \mathbf{0} \). (b) \( \nabla \|\mathbf{r}\| = \frac{\mathbf{r}^{*}}{\|\mathbf{r}\|} \).

Step 1: Define the Radius Vector

The radius vector \( \mathbf{r} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k} \) is given in the problem. It represents a vector in three-dimensional space with components \( x, y, \) and \( z \).

Step 2: Calculate the Curl of \( \mathbf{r} \)

The curl of a vector field \( \mathbf{F} = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k} \) is given by \( abla \times \mathbf{F} = \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} \right) \mathbf{i} + \left( \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x} \right) \mathbf{j} + \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \mathbf{k} \). For \( \mathbf{r} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k} \), take \( P = x, Q = y, R = z \). Substituting into the curl formula, each term is zero as the partial derivatives cancel each other out. Thus, \( abla \times \mathbf{r} = \mathbf{0} \).

Step 3: Find the Magnitude of \( \mathbf{r} \)

The magnitude of the vector \( \mathbf{r} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k} \) is \( \|\mathbf{r}\| = \sqrt{x^2 + y^2 + z^2} \).

Step 4: Compute the Gradient of the Magnitude of \( \mathbf{r} \)

The gradient of a scalar function \( f \) is defined as \( abla f = \frac{\partial f}{\partial x}\mathbf{i} + \frac{\partial f}{\partial y}\mathbf{j} + \frac{\partial f}{\partial z}\mathbf{k} \). For \( f = \|\mathbf{r}\| = \sqrt{x^2 + y^2 + z^2} \), compute each partial derivative: \( \frac{\partial f}{\partial x} = \frac{x}{\sqrt{x^2 + y^2 + z^2}} \), \( \frac{\partial f}{\partial y} = \frac{y}{\sqrt{x^2 + y^2 + z^2}} \), \( \frac{\partial f}{\partial z} = \frac{z}{\sqrt{x^2 + y^2 + z^2}} \). Hence, \( abla \|\mathbf{r}\| = \frac{x}{\|\mathbf{r}\|}\mathbf{i} + \frac{y}{\|\mathbf{r}\|}\mathbf{j} + \frac{z}{\|\mathbf{r}\|}\mathbf{k} \).

Step 5: Relate the Gradient to \( \mathbf{r}^{*} \)

Notice that \( abla \|\mathbf{r}\| = \frac{x}{\|\mathbf{r}\|}\mathbf{i} + \frac{y}{\|\mathbf{r}\|}\mathbf{j} + \frac{z}{\|\mathbf{r}\|}\mathbf{k} \) is the same as \( \frac{\mathbf{r}}{\|\mathbf{r}\|} \). Thus, it confirms that \( abla \|\mathbf{r}\| = \frac{\mathbf{r}^{*}}{\|\mathbf{r}\|} \).

Key Concepts

Understanding the Radius Vector

The radius vector is a fundamental concept in vector calculus. It acts as a position vector that points from the origin to any point in three-dimensional space. This vector is typically denoted as \( \mathbf{r} \) and expressed in terms of its components along the x, y, and z axes: \( \mathbf{r} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k} \).

• Components: Each component of \( \mathbf{r} \) corresponds to the coordinates \( x, y, \) and \( z \).
• Vector Notation: The unit vectors \( \mathbf{i}, \mathbf{j}, \mathbf{k} \) denote directions along the x-axis, y-axis, and z-axis respectively.

The radius vector is crucial because it helps identify a point's location in space in standard Cartesian coordinates. When visualized, it essentially forms a straight line segment from the origin to the coordinate point (x, y, z). Understanding this basic idea is vital for following subsequent vector operations like calculating the curl or the gradient.

Curl of a Vector Field

The curl is an operation applied to a vector field that measures its tendency to rotate around a point. One often calculated curl example is that of the radius vector \( \mathbf{r} \), which is given by \( abla \times \mathbf{r} \). The general formula for finding the curl of a vector field is: \[ abla \times \mathbf{F} = \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} \right) \mathbf{i} + \left( \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x} \right) \mathbf{j} + \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \mathbf{k} \]When this is applied to the radius vector \( \mathbf{r} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k} \), we find:

• Partial Derivatives: Calculate partial derivatives with respect to each variable, which in the radius vector's case results in each term cancelling out or becoming zero.
• Conclusion: Hence, \( abla \times \mathbf{r} = \mathbf{0} \), indicating that the radius vector does not have a rotational component at any point in space.

This means that the line connecting the origin to any point described by \( \mathbf{r} \) does not exhibit any swirling or rotating tendency at the point, reinforcing our understanding of it as a simple position vector.

Gradient of a Magnitude

The gradient is another key operation in vector calculus, used to determine the rate and direction of change in a scalar field. It is particularly useful for understanding how a function changes in multidimensional space.

For finding the gradient of the magnitude of the radius vector, we start with the formula for the magnitude \( \|\mathbf{r}\| = \sqrt{x^2 + y^2 + z^2} \) and recall the definition of a gradient \( abla f \) as:

\[ abla f = \frac{\partial f}{\partial x}\mathbf{i} + \frac{\partial f}{\partial y}\mathbf{j} + \frac{\partial f}{\partial z}\mathbf{k} \]Applying this to \( \|\mathbf{r}\| \):

• Partial Derivatives: For x, \( \frac{x}{\|\mathbf{r}\|} \); for y, \( \frac{y}{\|\mathbf{r}\|} \); for z, \( \frac{z}{\|\mathbf{r}\|} \).
• Gradient Formula: The gradient \( abla \|\mathbf{r}\| \) results in \( \frac{x}{\|\mathbf{r}\|}\mathbf{i} + \frac{y}{\|\mathbf{r}\|}\mathbf{j} + \frac{z}{\|\mathbf{r}\|}\mathbf{k} \),

This calculated gradient matches \( \frac{\mathbf{r}}{\|\mathbf{r}\|} \), a unit vector in the direction of \( \mathbf{r} \), showing how the radius vector magnifies into this standardized direction. It essentially underscores how the vector's magnitude flows outwards from the origin, consistent with the base direction identified by \( \mathbf{r} \).