Question

1. Consider vector field \textbf{F} = 2\textbf{i} + z^2\textbf{j} - y^2\textbf{k} and a family of loop curves \textbf{C} given by \textbf{r}(\phi) = \textbf{i} + a\cos\phi\textbf{j} + a\sin\phi\textbf{k}, \quad 0 \le \phi \le 2\pi \qquad \text{where } a > 0. (a) Show that \(\oint_C \textbf{F} \cdot d\textbf{r} = 0\) for all \(a > 0\). (b) Does the result in (a) imply that \textbf{F} is conservative? Why? Find \(\nabla \times \textbf{F}\)

          1. Consider vector field \textbf{F} = 2\textbf{i} + z^2\textbf{j} - y^2\textbf{k} and a family of loop curves \textbf{C} given by
\textbf{r}(\phi) = \textbf{i} + a\cos\phi\textbf{j} + a\sin\phi\textbf{k}, \quad 0 \le \phi \le 2\pi \qquad \text{where } a > 0.
(a) Show that \(\oint_C \textbf{F} \cdot d\textbf{r} = 0\) for all \(a > 0\).
(b) Does the result in (a) imply that \textbf{F} is conservative? Why? Find \(\nabla \times \textbf{F}\)

1. Consider vector field F = 2i + z^2j - y^2k and a family of loop curves C given by
r(ϕ) = i + acosϕj + asinϕk, 0 ≤ϕ≤2π where a > 0.
(a) Show that F· dr = 0 for all a > 0.
(b) Does the result in (a) imply that F is conservative? Why? Find ∇×F

Added by Wayne M.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Shaiju T

Step 1

The line integral of F along C is given by: ∮ F · dr = ∫ F · dr = ∫ (2i + z^2j - y^2k) · (dx/dt i + dy/dt j + dz/dt k) dt Substituting the parametric equations of C into the line integral, we have: ∮ F · dr = ∫ (2i + (asin(t))^2j - (acos(t))^2k) · (-asin(t) i + Show more…

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Consider the vector field F = 2i + z^2j - y^2k and a family of loop curves C given by r(t) = i + acos(t)j + asin(t)k, 0 â‰¤ t â‰¤ 2Ï€, where a > 0. (a) Show that âˆ® F Â· dr = 0 for all a > 0. (b) Does the result in (a) imply that F is conservative? Why? Find VF.