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Chapter 13: Problem 4

Determine whether or not the vector field is conservative. $$ \mathbf{F}(x, y)=15 x^{2} y^{2} \mathbf{i}+10 x^{3} y \mathbf{j} $$

Short Answer

Expert verified
Yes, the given vector field is conservative as the curl of the vector field is zero, indicated by the equality of partial derivatives.

Step by step solution

01

Compute partial derivatives

The first step is to calculate the partial derivative of \( F_{i} \) with respect to y and \( F_{j} \) with respect to x. Here \( F_{i} = 15x^{2}y^{2} = P \) and \( F_{j} = 10x^{3}y = Q \). So, \( \frac{\partial P}{\partial y} = 30x^{2}y \) and \( \frac{\partial Q}{\partial x} = 30x^{2}y \).
02

Check if the vector field is conservative

The crucial step in determining if a vector field is conservative is to check if the curl of the vector field is zero. In other words, if \( \frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x} \). As both our computed partial derivatives are equal ( \( \frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x} = 30x^{2}y \)), this verifies that the given vector field is indeed conservative.

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Most popular questions from this chapter

Evaluate \(\int_{C} \mathbf{F} \cdot d \mathbf{r}\) where \(C\) is represented by \(\mathbf{r}(t)\) \(\mathbf{F}(x, y)=x y \mathbf{i}+y \mathbf{j}\) \(\quad C: \mathbf{r}(t)=4 \cos t \mathbf{i}+4 \sin t \mathbf{j}, \quad 0 \leq t \leq \pi / 2\)

Find the divergence of the vector field \(\mathrm{F}\). \(\mathbf{F}(x, y, z)=\ln \left(x^{2}+y^{2}\right) \mathbf{i}+x y \mathbf{j}+\ln \left(y^{2}+z^{2}\right) \mathbf{k}\)

Demonstrate the property that \(\int_{C} \mathbf{F} \cdot d \mathbf{r}=\mathbf{0}\) regardless of the initial and terminal points of \(C,\) if the tangent vector \(\mathbf{r}^{\prime}(t)\) is orthogonal to the force field \(\mathbf{F}\) \(\mathbf{F}(x, y)=x \mathbf{i}+y \mathbf{j}\) \(C: \mathbf{r}(t)=3 \sin t \mathbf{i}+3 \cos t \mathbf{j}\)

Use a computer algebra system to evaluate the integral \(\int_{C} \mathbf{F} \cdot d \mathbf{r}\) where \(C\) is represented by \(\mathbf{r}(t)\) \(\mathbf{F}(x, y, z)=\frac{x \mathbf{i}+y \mathbf{j}+z \mathbf{k}}{\sqrt{x^{2}+y^{2}+z^{2}}}\) \(C: \mathbf{r}(t)=t \mathbf{i}+t \mathbf{j}+e^{t} \mathbf{k}, \quad 0 \leq t \leq 2\)

Prove the property for vector fields \(\mathbf{F}\) and \(\mathbf{G}\) and scalar function \(f .\) (Assume that the required partial derivatives are continuous.) $$ \operatorname{div}(\mathbf{F}+\mathbf{G})=\operatorname{div} \mathbf{F}+\operatorname{div} \mathbf{G} $$
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