Question

a. Prove that the rotation field $\mathbf{F}=\frac{⟨-y,x⟩}{|\mathbf{r}{|}^p},$ where $\mathbf{r}=⟨x,y⟩$ is not conservative for $p\ne 2$ b. For $p=2,$ show that $\mathbf{F}$ is conservative on any region not containing the origin. c. Find a potential function for $\mathbf{F}$ when $p=2$

Short answer

No, the given vector field is not conservative for $p\ne 2$. 2. What is the curl of the vector field when $p=2$? The curl of the vector field when $p=2$ is $\mathrm{\nabla }\times \mathbf{F}=\left(\frac{x^2-y^2}{(|\mathbf{r}{|}^2{)}^2}\right)\hat{k}$. 3. Is the vector field conservative in any region when $p=2$? Yes, the vector field is conservative in any region that doesn't include the origin when $p=2$. 4. What is the potential function for the vector field when $p=2$? The potential function for the vector field when $p=2$ is $f(x,y)=-\frac{xy}{|\mathbf{r}{|}^2}+c$, where c is a constant.

Step-by-step solution

Finding the Curl of the Vector Field

Let's first find the curl of given vector field $\mathbf{F}$. Recall the formula for the curl of a 2D vector field: $\mathrm{\nabla }\times \mathbf{F}=(\frac{\partial F_2}{\partial x}-\frac{\partial F_1}{\partial y})\hat{k}$. Here, $F_1=-\frac{y}{|\mathbf{r}{|}^p}$ and $F_2=\frac{x}{|\mathbf{r}{|}^p}$. Now, we will compute the partial derivatives and then the curl: $$\frac{\partial F_1}{\partial y}=\frac{\partial }{\partial y}(-\frac{y}{|\mathbf{r}{|}^p})=\frac{y^2-(x^2+y^2)|\mathbf{r}{|}^p}{(|\mathbf{r}{|}^p{)}^2}$$ $$\frac{\partial F_2}{\partial x}=\frac{\partial }{\partial x}\left(\frac{x}{|\mathbf{r}{|}^p}\right)=\frac{x^2-(x^2+y^2)|\mathbf{r}{|}^p}{(|\mathbf{r}{|}^p{)}^2}$$ Then, the curl is: $$\mathrm{\nabla }\times \mathbf{F}=(\frac{x^2-(x^2+y^2)|\mathbf{r}{|}^p}{(|\mathbf{r}{|}^p{)}^2}-\frac{y^2-(x^2+y^2)|\mathbf{r}{|}^p}{(|\mathbf{r}{|}^p{)}^2})\hat{k}$$

Proving the Vector Field is not Conservative for $p\ne 2$

Now, let's simplify the expression we obtained for the curl: $$\mathrm{\nabla }\times \mathbf{F}=\left(\frac{x^2-y^2}{(|\mathbf{r}{|}^p{)}^2}\right)\hat{k}$$ To show that the vector field is not conservative for $p\ne 2$, we just need to prove that $\mathrm{\nabla }\times \mathbf{F}\ne 0$. From the expression we derived, we can see that the curl will be zero if and only if $p=2$. Therefore, we have shown that the vector field is not conservative for $p\ne 2$.

Proving the Vector Field is Conservative for $p=2$

When $p=2$, the curl of the vector field becomes: $$\mathrm{\nabla }\times \mathbf{F}=\left(\frac{x^2-y^2}{(|\mathbf{r}{|}^2{)}^2}\right)\hat{k}$$ Given that the region doesn't contain the origin, we can see that the curl is indeed zero, which means the vector field is conservative for $p=2$.

Finding a Potential Function for $p=2$

Since we know that the vector field is conservative for $p=2$, we can find a potential function for it. Recall that if a conservative field $\mathbf{F}$ has a potential function $f$, it should satisfy the following: $$F_1=\frac{\partial f}{\partial x}\text{and}{F}_2=\frac{\partial f}{\partial y}$$ For $p=2$, we have $F_1=-\frac{y}{|\mathbf{r}{|}^2}$ and $F_2=\frac{x}{|\mathbf{r}{|}^2}$. Integrating $F_1$ with respect to $x$ gives: $$f(x,y)=-\frac{xy}{|\mathbf{r}{|}^2}+g(y)$$ Now, differentiate the function $f(x,y)$ with respect to $y$ and compare it with $F_2$: $$\frac{\partial f}{\partial y}=\frac{-x|\mathbf{r}{|}^2+2x{y}^2}{|\mathbf{r}{|}^4}+g^{\prime }(y)$$ Comparing this with $F_2=\frac{x}{|\mathbf{r}{|}^2}$, we can deduce that $g^{\prime }(y)=0$ and thus, $g(y)$ is a constant. Let's call this constant $c$. Then, the potential function for the vector field is given by: $$f(x,y)=-\frac{xy}{|\mathbf{r}{|}^2}+c$$

Key concepts

Curl

The concept of the curl of a vector field is like a measure of the field's rotation or swirling at a point. Think of the water swirling around a drain; that's a similar action to a non-zero curl in a vector field. For a two-dimensional vector field, the curl can be calculated using the formula: $$abla\times \mathbf{F}=(\frac{\partial F_2}{\partial x}-\frac{\partial F_1}{\partial y})\hat{k}$$This formula helps us determine whether a field is rotating around a point or not. A zero curl indicates that the field does not exhibit any rotational behavior - it's like calm water. Conversely, a non-zero curl suggests a swirling effect exists.

• Calculating the curl involves partial derivatives, as it's the rate of change in certain directions.
• The operator $\hat{k}$ denotes that the curl is behaving in the out-of-plane direction for two-dimensional fields.

The curl is a compelling tool in vector calculus because it tells us about the inherent rotational characteristics of a field.

Conservative Field

In mathematics and physics, a conservative field is a special type of vector field. It is one where the work done by the field on a particle moving from one point to another is path-independent. This means that the total work depends only on the start and end points, and not on the path taken between them. A field is conservative if and only if its curl is zero everywhere in the region of interest. Key characteristics of a conservative field include:

• Path independence: Different paths between two points require the same amount of work.
• Existence of a potential function: There exists a scalar potential function where the field is the gradient of this function.

These fields commonly arise in physical situations, such as gravity and electrostatics. Understanding if a field is conservative can greatly simplify analyses of physical systems.

Potential Function

A potential function is a scalar function whose gradient yields the vector field. It essentially encapsulates all the information of a conservative vector field into a simpler, scalar form. If a vector field $\mathbf{F}$ is conservative, there exists a potential function $f$ such that:$$\mathbf{F}=ablaf$$This relationship ensures that the components of the vector field are the partial derivatives of the potential function:

• $F_1=\frac{\partial f}{\partial x}$
• $F_2=\frac{\partial f}{\partial y}$

Finding a potential function involves reversing this gradient process, often by integrating each component of the vector field. The existence of a potential function is pivotal because it simplifies the analysis, converting problems involving vector fields into scalar ones.

Partial Derivatives

Partial derivatives are a cornerstone of multivariable calculus. They measure how a function changes as its input variables change, but only one variable at a time, keeping others constant. In vector fields, partial derivatives help calculate components such as the curl and gradient, which reveal information about the field's behavior.For example, consider a function $f(x,y)$.

• The partial derivative of $f$ with respect to $x$, denoted by $\frac{\partial f}{\partial x}$, tells us how $f$ changes as $x$ changes while $y$ remains constant.
• Similarly, $\frac{\partial f}{\partial y}$ indicates how $f$ changes as $y$ changes while $x$ remains constant.

These derivatives are not only fundamental in computing curls and assessing vector fields but also serve in optimization, slope approximation, and analyzing rates of change. Mastery of partial derivatives enriches one's ability to tackle a broad array of mathematical problems.