Unknown Vector Field

Calculus Level 5

Let $S$ be the subset of $\mathbb{R}^3$ consisting of the union of:

$\text{i})$ the $z-$ axis,

$\text{ii})$ the unit circle $x^2+y^2 = 1, \space \ z = 0$ ,

$\text{iii})$ the points $(0,y,0)$ with $y \ge 1.$

Let $A$ be the open set $\mathbb{R}^3-S$ of $\mathbb{R}^3.$ Let $C_1, C_2, D_1, D_2, D_3$ be the oriented curves in $A$ that are pictured below. Suppose $\bf{F}$ is a vector field on $A$ , with $\text{Curl} \space \bf{F} = 0$ , and that

$\displaystyle \int_{C_1} \textbf{F} \cdot \text d \textbf{r} = 3 \ \text{and} \ \int_{C_2} \textbf{F} \cdot \text d \textbf{r} = 7$

Find the value of $\displaystyle \sum_i \int_{D_i} \textbf{F} \cdot \text d \textbf{r}.$

The answer is 10.

1 solution

Akeel Howell
Apr 19, 2018

Since we are not given a surface, we must build one with the given constraints (note that we don't need to explicitly know the vector field since its curl is $\textbf{0}$ ):

In the picture above, dotted curves represent parts of the set $S$ contained within the surface, or parts of the surface that are formed behind other parts of the manifold. Grey shadings represent the surface, and the curves are its boundary curves.

Stoke's Theorem states that $\displaystyle \oint_{\partial D} \textbf{F} \cdot \text{d} \textbf{s} = \iint_D \text{Curl} \space \textbf{F} \cdot \text d \textbf{S}$ .

For boundary curves $D_2$ and $D_3$ , which are oriented such that their line integrals cancel, this is $0$ .

Notice how $D_1$ wraps around the $z-$ axis and the unit circle in the picture. Given the orientation of this boundary curve, we have that $\displaystyle \int_{D_1} \textbf{F} \cdot \text{d} \textbf{r} = \int_{C_1+C_2} \textbf{F} \cdot \text d \textbf{r} = 3+7 = 10.$

With that, we have $\displaystyle \sum_i \int_{D_i} \textbf{F} \cdot \text d \textbf{r} = \boxed{10}.$

Unknown Vector Field

The answer is 10.

1 solution

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