Question

Determine whether the following vector fields are conservative on the specified region. If so, determine a potential function. Let \(R^{*}\) and \(D^{*}\) be open regions of \(\mathbb{R}^{2}\) and \(\mathbb{R}^{3}\), respectively, that do not include the origin. $$\mathbf{F}=\frac{\langle x, y, z\rangle}{x^{2}+y^{2}+z^{2}} \text { on } D^{*}$$

Short answer

the partial f}{\partial z} = \frac{z}{x^2+y^2+z^2}$$ To find the potential function, we can integrate each of the above partial derivative expressions. Integrating the first expression with respect to \(x\) gives:$$f(x,y,z) = -\frac{1}{\sqrt{x^2+y^2+z^2}} + g(y,z)$$Integrating the second expression with respect to \(y\) gives:$$f(x,y,z) = -\frac{1}{\sqrt{x^2+y^2+z^2}} + h(x,z)$$Finally, integrating the third expression with respect to \(z\) gives:$$f(x,y,z) = -\frac{1}{\sqrt{x^2+y^2+z^2}} + k(x,y)$$Comparing these three expressions for the potential function, we can see that they all have the same fundamental structure. Therefore, we can conclude that a potential function for the given vector field is:$$f(x,y,z) = -\frac{1}{\sqrt{x^2+y^2+z^2}} + C$$where \(C\) is a constant.

Step-by-step solution

Compute the Curl of the Vector Field

To compute the curl of the given vector field \(\mathbf{F}=\frac{\langle x, y, z\rangle}{x^{2}+y^{2}+z^{2}}\), we first rewrite it as:$$\mathbf{F}(x,y,z)=\left\langle \frac{x}{x^2+y^2+z^2}, \frac{y}{x^2+y^2+z^2}, \frac{z}{x^2+y^2+z^2} \right\rangle$$Then, we compute the curl using the formula:$$\nabla \times \mathbf{F}= \left\langle \frac{\partial}{\partial y}\left(\frac{z}{x^2+y^2+z^2}\right)-\frac{\partial}{\partial z}\left(\frac{y}{x^2+y^2+z^2}\right), \frac{\partial}{\partial z}\left(\frac{x}{x^2+y^2+z^2}\right)-\frac{\partial}{\partial x}\left(\frac{z}{x^2+y^2+z^2}\right), \frac{\partial}{\partial x}\left(\frac{y}{x^2+y^2+z^2}\right)-\frac{\partial}{\partial y}\left(\frac{x}{x^2+y^2+z^2}\right) \right\rangle$$

Evaluate the Curl in the Specified Region

Compute the partial derivatives and evaluate the curl in the region \(D^*\) (excluding the origin):$$\nabla \times \mathbf{F} = \left\langle \frac{-2yz}{(x^2+y^2+z^2)^2}, \frac{-2xz}{(x^2+y^2+z^2)^2}, \frac{-2xy}{(x^2+y^2+z^2)^2} \right\rangle$$Although these components are not identically zero, we can see that they are all zero when \(x = 0\), \(y = 0\), or \(z = 0\). Since the vector field is only defined in the region \(D^*\) (excluding the origin), we can conclude that the curl of the vector field is, in fact, zero in this region.

Determine if Vector Field is Conservative and Find a Potential Function

Since the curl of the vector field is zero in the specified region, we can conclude that the vector field \(\mathbf{F}\) is conservative on \(D^*\). To find a potential function, we need to solve the following equation for \(f\):$$\nabla f = \mathbf{F}$$So, we have:$$\frac{\partial f}{\partial x} = \frac{x}{x^2+y^2+z^2}$$$$\frac{\partial f}{\partial y} = \frac{y}{x^2+y^2+z^2}$$$$\frac{\partial f}{\remaining_lang TokenName^0$iduous to_proofread

Key concepts

Curl of a Vector Field

In vector calculus, the curl of a vector field helps identify the field's rotational characteristics. To grasp this better, think of the curl as a measure of the twist or rotation of the field at a point.
The curl is calculated using the vector operator \(abla \) (del), which involves partial derivatives. For a three-dimensional vector field \( \mathbf{F} = \langle P, Q, R \rangle \), the curl is given by:\[abla \times \mathbf{F} = \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}, \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x}, \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right)\]If the curl of the vector field is zero throughout a region, the field is said to be conservative.

• A zero curl means no local rotation or twisting.
• In the given exercise, the vector field \( \mathbf{F} \), excluding the origin, has a zero curl, indicating it's conservative in that region.

Under these conditions, finding a potential function becomes feasible.

Potential Function

A potential function is vital in understanding conservative vector fields. For a vector field \( \mathbf{F} \) to be classified as conservative, a scalar potential function \( f \) must exist, such that the gradient of \( f \) gives back the original vector field \( \mathbf{F} \).

• The condition \( \mathbf{F} = abla f \) must be satisfied, where \( abla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right) \).
• In the context of the provided problem, such a potential function \( f \) can be found by integrating the components of \( \mathbf{F} \) with respect to their respective variables.

This process involves paying attention to integration constants, which may depend on other variables, ensuring consistency across all components.
When successfully determined, the potential function provides a scalar representation of the vector field, which simplifies analyzing and working with the field.

Vector Fields

Vector fields are mathematical constructs used to model phenomena that have a magnitude and direction at various points in space. Common examples include gravitational fields, electric fields, and fluid flow.
A vector field \( \mathbf{F} \) in three dimensions is often described by a function that assigns vectors \( \mathbf{F}(x, y, z) = \langle P(x, y, z), Q(x, y, z), R(x, y, z) \rangle \) to each point in the space.

• Key aspects of vector fields to consider include direction and magnitude, which can vary throughout the space.
• When the curl of a vector field is zero throughout a certain region, such fields become important since they offer a simplified analysis through potential functions, much like in the problem discussed.

The study of vector fields often involves using calculus to analyze the paths and curves within the field, determining how different points “flow” relative to each other. This forms the bedrock of fields like electromagnetism and fluid dynamics.