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Chapter 13: Problem 15

Verify that the vector field is conservative. \(\mathbf{F}(x, y)=12 x y \mathbf{i}+6\left(x^{2}+y\right) \mathbf{j}\)

Short Answer

Expert verified
Yes, the given vector field is conservative as the curl of F is zero.

Step by step solution

01

Identifying the Vector Components

First, it's important to assign each part of the vector field to its corresponding component in two dimensions, \(x\) and \(y\). So, \(F_{x} = 12xy\) and \(F_{y} = 6x^2 + 6y\).
02

Verifying the Conservative Condition

Next, a vector field is conservative if and only if the curl of F is zero, which means the partial derivative of \(F_y\) with respect to \(x\) equals the partial derivative of \(F_x\) with respect to \(y\). Let's calculate these partial derivatives.
03

Calculate the Partial Derivatives

The partial derivative of \(F_y\) with respect to \(x\) is \(\frac{\partial F_y}{\partial x} = \frac{\partial}{\partial x} (6x^2 + 6y) = 12x\). The partial derivative of \(F_x\) with respect to \(y\) is \(\frac{\partial F_x}{\partial y} = \frac{\partial}{\partial y} (12xy) = 12x\). It can be observed that \(\frac{\partial F_y}{\partial x} = \frac{\partial F_x}{\partial y}\).
04

Final Analysis

Since the curl of F is zero, as \(\frac{\partial F_y}{\partial x} = \frac{\partial F_x}{\partial y}\), we can conclude that the given vector field F is conservative.

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Most popular questions from this chapter

In Exercises \(25-30,\) evaluate \(\int_{C} \mathbf{F} \cdot d \mathbf{r}\) where \(C\) is represented by \(\mathbf{r}(t)\) \(\mathbf{F}(x, y)=x y \mathbf{i}+y \mathbf{j}\) \(\quad C: \mathbf{r}(t)=4 t \mathbf{i}+t \mathbf{j}, \quad 0 \leq t \leq 1\)

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 \(\mathbf{F}\) \(\mathbf{F}(x, y)=-3 y \mathbf{i}+x \mathbf{j}\) \(C: \mathbf{r}(t)=t \mathbf{i}-t^{3} \mathbf{j}\)

Find the total mass of two turns of a spring with density \(\rho\) in the shape of the circular helix \(\mathbf{r}(t)=3 \cos t \mathbf{i}+3 \sin t \mathbf{j}+2 t \mathbf{k}\) \(\rho(x, y, z)=z\)

Evaluate \(\int_{C}\left(2 x+y^{2}-z\right) d s\) along the given path. \(C:\) line segments from (0,0,0) to (0,1,0) to (0,1,1) to (0,0,0)

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