Question

Consider the vector field \(\underline{v}(\underline{x}) = (y^2 + z^2, x^2 + z^2, x^2 + y^2)\). (a) Show that \((\nabla \times \underline{v}) \cdot \underline{x} = 0. (b) Calculate \(\int_C \underline{v} \cdot d\underline{r}\), where \(C\) is the curve defined by \(x^2 + z^2 = 4\), \(y = 0\), \(z \ge 0\) from \((2, 0, 0)\) to \((-2, 0, 0)\). (c) Let \(\Gamma\) be any curve from \((2, 0, 0)\) to \((-2, 0, 0)\) on the sphere \(x^2 + y^2 + z^2 = 4\). Use the results from (a), (b) and Stokes' Theorem to calculate \(\int_\Gamma \underline{v} \cdot d\underline{r}\)

          Consider the vector field \(\underline{v}(\underline{x}) = (y^2 + z^2, x^2 + z^2, x^2 + y^2)\).
(a) Show that \((\nabla \times \underline{v}) \cdot \underline{x} = 0.
(b) Calculate \(\int_C \underline{v} \cdot d\underline{r}\), where \(C\) is the curve defined by \(x^2 + z^2 = 4\), \(y = 0\), \(z \ge 0\)
from \((2, 0, 0)\) to \((-2, 0, 0)\).
(c) Let \(\Gamma\) be any curve from \((2, 0, 0)\) to \((-2, 0, 0)\) on the sphere \(x^2 + y^2 + z^2 = 4\).
Use the results from (a), (b) and Stokes' Theorem to calculate \(\int_\Gamma \underline{v} \cdot d\underline{r}\)

Consider the vector field v(x) = (y^2 + z^2, x^2 + z^2, x^2 + y^2).
(a) Show that (∇×v) ·x = 0.
(b) Calculate v· dr, where C is the curve defined by x^2 + z^2 = 4, y = 0, z ≥ 0
from (2, 0, 0) to (-2, 0, 0).
(c) Let Γ be any curve from (2, 0, 0) to (-2, 0, 0) on the sphere x^2 + y^2 + z^2 = 4.
Use the results from (a), (b) and Stokes' Theorem to calculate ∫Γv· dr

Added by Joshua V.

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