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Chapter 14: Problem 3

How do you determine whether a vector field in \(\mathbb{R}^{2}\) is conservative (has a potential function \(\varphi\) such that \(\mathbf{F}=\nabla \varphi\) )?

Short Answer

Expert verified
Question: Determine whether the following vector field is conservative and find its potential function, if it exists: \(\mathbf{F} = (3x^2 + 2y, -2x + 4y^3)\). Answer: Following the proposed steps: Step 1: Identify the vector field components In this case, we have \(P(x, y) = 3x^2 + 2y\) and \(Q(x, y) = -2x + 4y^3\). Step 2: Compute the partial derivatives We have: $$\frac{\partial P}{\partial y} = 2$$ And: $$\frac{\partial Q}{\partial x} = -2$$ Step 3: Check if the condition holds Here, we see that: $$\frac{\partial P}{\partial y} \neq \frac{\partial Q}{\partial x}$$ Since the condition doesn't hold, the vector field is not conservative, and therefore, there is no potential function.

Step by step solution

01

Identify the vector field components

Given a vector field \(\mathbf{F} = (P(x, y), Q(x, y))\) in \(\mathbb{R}^{2}\), identify the components P and Q.
02

Compute the partial derivatives

Calculate the partial derivatives of P with respect to y and Q with respect to x: $$\frac{\partial P}{\partial y} $$ and $$\frac{\partial Q}{\partial x} $$
03

Check if the condition holds

Compare the computed partial derivatives from step 2: $$\frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x}$$ If this condition holds, the vector field is conservative. If they are not equal, the vector field is not conservative.
04

Find the potential function (optional)

If the vector field is conservative and you want to find the potential function \(\varphi(x, y)\), use the following method: 4.1. Integrate P with respect to x: $$\varphi_1(x, y) = \int P dx$$ Keep in mind that you will get a constant of integration that may depend on y. 4.2. Integrate Q with respect to y: $$\varphi_2(x, y) = \int Q dy$$ Keep in mind that you will get a constant of integration that may depend on x. 4.3. Combine the results of 4.1 and 4.2 to find the overall potential function \(\varphi(x, y)\). Make sure to use the constants of integration to ensure consistency between both results. Verify that the gradients of the potential function match with the original vector field components.

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Most popular questions from this chapter

Find the work required to move an object in the following force fields along a line segment between the given points. Check to see whether the force is conservative. $$\mathbf{F}=\langle x, y\rangle \text { from } A(1,1) \text { to } B(3,-6)$$

a. For what values of \(a, b, c,\) and \(d\) is the field \(\mathbf{F}=\langle a x+b y, c x+d y\rangle\) conservative? b. For what values of \(a, b,\) and \(c\) is the field \(\mathbf{F}=\left\langle a x^{2}-b y^{2}, c x y\right\rangle\) conservative?

a. Prove that the rotation field \(\mathbf{F}=\frac{\langle-y, x\rangle}{|\mathbf{r}|^{p}},\) where \(\mathbf{r}=\langle x, y\rangle\) is not conservative for \(p \neq 2\) 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 \(p=2\)

Use the procedure in Exercise 57 to construct potential functions for the following fields. $$\mathbf{F}=\left\langle 2 x^{3}+x y^{2}, 2 y^{3}+x^{2} y\right\rangle$$

Prove the following identities. a. \(\iiint_{D} \nabla \times \mathbf{F} d V=\iint_{S}(\mathbf{n} \times \mathbf{F}) d S\) (Hint: Apply the Divergence Theorem to each component of the identity.) b. \(\iint_{S}(\mathbf{n} \times \nabla \varphi) d S=\oint_{C} \varphi d \mathbf{r}\) (Hint: Apply Stokes' Theorem to each component of the identity.)
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