Is the Parameterization Correct in Leithold's Stokes' Theorem Problem?

runinfang · Mar 20, 2024

The question is a problem from Leithold's calculus book. I didn't understand the (x = 5 \cos(t)). Shouldn't it be (x = 2 cos(t))? I'm referring to item b.

Handwritten-style math worked solution, problem 30: F(x,y) = y^2 i + x^2 j over the curve z = 5 cos t, y = 2 sin t, 0 ≤ t ≤ 2π; parts (a) and (b) with line-integral and Green's theorem steps.

i tried this way. i don't know what is wrong.

Worked solution in plain text: circle x^2 + y^2 = 4 parametrized as x = 2cos(t), y = 2sin(t) for 0 ≤ t ≤ 2π; unit tangent T(t) = (−sin(t), cos(t)); line integral of (−8sin^2(t) + 8cos^2(t)) dt = 0; surface integral of (rot F)·k dA = 0 since rot F = 0.

WWGD · Mar 20, 2024

runinfang said:

Homework Statement: Verify Stokes' theorem in the plane where f(x,y)=y2i+x2j
and the region formed is bounded by the circle (x^2+y^2=4).
Relevant Equations: Stokes' theorem in the plane
"Let \( M \) and \( N \) be the functions, \( C \) be the curve, and \( R \) be the region defined as in Green's theorem. If \( \mathbf{F}(x, y) = M(x, y)\mathbf{i} + N(x, y)\mathbf{j} \) and \( \mathbf{T}(s) \) is the unit tangent vector to \( C \) at \( P \), where \( s \) units is the arc length of \( C \) measured from a point \( P_0 \) to \( P \), then:

\[
\oint_C \mathbf{F} \cdot \mathbf{T} \, ds = \iint_R (\text{curl } \mathbf{F}) \cdot \mathbf{k} \, dA
\]"

The question is a problem from Leithold's calculus book. I didn't understand the (x = 5 \cos(t)). Shouldn't it be (x = 2 cos(t))? I'm referring to item b.
View attachment 342076
i tried this way. i don't know what is wrong.
View attachment 342077

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Is the Parameterization Correct in Leithold's Stokes' Theorem Problem?

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