Question

In Exercises \(17-20\), a conservative vector field \(\vec{F}\) and a curve C are given. 1\. Find a potential function \(f\) for \(\vec{F}\) 2\. Compute curl \(\vec{F}\). 3\. Evaluate \(\int_{C} \vec{F} \cdot d \vec{r}\) directly, i.e., using Key Idea 14.3.1. 4\. Evaluate \(\int_{C} \vec{F} \cdot d \vec{r}\) using the Fundamental Theorem of Line Integrals. $$ \begin{aligned} &\vec{F}=\langle 2 x, 2 y, 2 z\rangle, C \text { is curve parametrized by } \vec{r}(t)=\\\ &\langle\cos t, \sin t, \sin (2 t)\rangle \text { on } 0 \leq t \leq 2 \pi \end{aligned} $$

Short answer

The line integral \(\int_{C} \vec{F} \cdot d\vec{r}\) is 0.

Step-by-step solution

Identify Components of the Vector Field

Given the vector field \(\vec{F} = \langle 2x, 2y, 2z \rangle\), identify the components: \(F_1 = 2x\), \(F_2 = 2y\), and \(F_3 = 2z\).

Find the Potential Function

Since \(\vec{F}\) is conservative, there exists a potential function \(f\) such that \(abla f = \vec{F}\). Find \(f\) by integrating each component: 1. Integrate \(2x\) with respect to \(x\) to get \(x^2 + g(y, z)\).2. Integrate \(2y\) with respect to \(y\) to get \(y^2 + h(x, z)\).3. Integrate \(2z\) with respect to \(z\) to get \(z^2 + k(x, y)\).Combine to get \(f(x, y, z) = x^2 + y^2 + z^2\), considering constant terms appropriately.

Compute the Curl of \(\vec{F}\)

The curl of a vector field \(\vec{F} = \langle F_1, F_2, F_3 \rangle\) is defined as \(abla \times \vec{F}\). Compute each component as follows:\(abla \times \vec{F} = \left(\frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z}, \frac{\partial F_1}{\partial z} - \frac{\partial F_3}{\partial x}, \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y}\right)\).For \(\vec{F} = \langle 2x, 2y, 2z \rangle\), all partial derivatives involved in the curl are zero, so \(abla \times \vec{F} = \langle 0, 0, 0 \rangle\).

Parametrize the Curve and Compute the Line Integral Directly

The curve \(C\) is parametrized by \(\vec{r}(t) = \langle \cos t, \sin t, \sin(2t) \rangle\) for \(0 \leq t \leq 2\pi\). Compute the integral \(\int_{C} \vec{F} \cdot d\vec{r}\):1. Calculate \(\vec{F}(\vec{r}(t)) = \langle 2\cos t, 2\sin t, 2\sin(2t) \rangle\).2. Compute \(d\vec{r}/dt = \langle -\sin t, \cos t, 2\cos(2t) \rangle\).3. Compute the dot product: \(\vec{F} \cdot \frac{d\vec{r}}{dt} = 2(-\cos t\sin t + \sin t \cos t + \sin(2t)\cdot 2\cos(2t)) = 0\) due to trigonometric identities over one full period.4. Evaluate the integral: \(\int_{0}^{2\pi} 0\; dt = 0\).

Use the Fundamental Theorem of Line Integrals

According to the Fundamental Theorem of Line Integrals, if \(\vec{F} = abla f\) for some potential function \(f\), then \(\int_{C} \vec{F} \cdot d\vec{r} = f(\vec{r}(b)) - f(\vec{r}(a))\), where \(a\) and \(b\) are the endpoints of \(C\).1. Calculate \(f(\vec{r}(0)) = f(\langle 1, 0, 0 \rangle) = 1^2 + 0^2 + 0^2 = 1\).2. Calculate \(f(\vec{r}(2\pi)) = f(\langle 1, 0, 0 \rangle) = 1^2 + 0^2 + 0^2 = 1\).3. Use the theorem to find \(\int_{C} \vec{F} \cdot d\vec{r} = 1 - 1 = 0\).

Key concepts

Potential Function

A potential function is a scalar function whose gradient is a given vector field. For conservative vector fields, like the one described by \(\vec{F} = \langle 2x, 2y, 2z \rangle\), a potential function \(f\) exists such that \(abla f = \vec{F}\). This means applying the gradient to \(f\) will yield the original vector field.

To find the potential function, we integrate each component of the vector field. For the vector field components \(2x\), \(2y\), and \(2z\), you integrate each component with respect to its respective variable:

• Integrate \(2x\) with respect to \(x\) to get \(x^2 + g(y, z)\).
• Integrate \(2y\) with respect to \(y\) to get \(y^2 + h(x, z)\).
• Integrate \(2z\) with respect to \(z\) to obtain \(z^2 + k(x, y)\).

Combine these results to form the potential function \(f(x, y, z) = x^2 + y^2 + z^2\). The functions \(g, h,\) and \(k\) account for zero because no other terms are introduced across the integrations.

Curl of a Vector Field

The curl of a vector field is a measure of the rotation at a point in the field. It is defined mathematically as \(abla \times \vec{F}\) for a vector field \(\vec{F} = \langle F_1, F_2, F_3 \rangle\). Essentially, it tells you how much the field is "twisting" at a point.

For the given vector field \(\vec{F} = \langle 2x, 2y, 2z \rangle\), you compute each component of the curl:

• The \(x\)-component: \(\frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z} = 0\) since both derivatives are zero.
• The \(y\)-component: \(\frac{\partial F_1}{\partial z} - \frac{\partial F_3}{\partial x} = 0\).
• The \(z\)-component: \(\frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y} = 0\).

Thus, the curl of \(\vec{F}\) is \(\langle 0, 0, 0 \rangle\), indicating no rotation, confirming that the vector field is indeed conservative.

Line Integral

The line integral of a vector field along a curve measures the field's influence over the trajectory of that curve. It is evaluated by integrating the dot product of the field and the curve's derivative over the curve's parameterization.

Given a curve \(C\) parameterized by \(\vec{r}(t) = \langle \cos t, \sin t, \sin(2t) \rangle\) for \(0 \leq t \leq 2\pi\), and a vector field \(\vec{F} = \langle 2\cos t, 2\sin t, 2\sin(2t) \rangle\), the steps are:

• Compute the derivative of \(\vec{r}(t)\), \(\frac{d\vec{r}}{dt} = \langle -\sin t, \cos t, 2\cos(2t) \rangle\).
• Calculate the dot product \(\vec{F} \cdot \frac{d\vec{r}}{dt}\).
• Integrating over \(t\) from \(0\) to \(2\pi\) results in \(0\), due to the periodic nature of the trigonometric functions.

This result indicates that the net "flow" across the given closed path is zero, which aligns with the property of conservative fields.

Fundamental Theorem of Line Integrals

The Fundamental Theorem of Line Integrals is a powerful tool for simplifying the evaluation of line integrals when dealing with a conservative vector field. It states that if a vector field is the gradient of a potential function, then the line integral between two points depends solely on the values of the potential function at those points.

For a conservative vector field \(\vec{F} = abla f\), the theorem asserts:
\[\int_{C} \vec{F} \cdot d\vec{r} = f(\vec{r}(b)) - f(\vec{r}(a))\]
Given the curve \(C\) between the points \(\vec{r}(0) = \langle 1, 0, 0 \rangle\) and \(\vec{r}(2\pi) = \langle 1, 0, 0 \rangle\), calculate the potential function \(f\) at both points:

• \(f(\vec{r}(0)) = 1\)
• \(f(\vec{r}(2\pi)) = 1\)

The difference of potential values gives zero, \(1 - 1 = 0\), confirming the result of the direct line integral calculation. This shows the power of the theorem in simplifying such evaluations.