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Let C' be the top half of the circle x² + y² = 16, and let F(x, y) = ( − y, x). So Then F.dr =

Let C' be the top half of the circle x² + y² = 16, and let F(x, y) = ( − y, x). So Then F.dr =

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Let \( C \) be the top half of the circle \( x^2 + y^2 = 16 \), and let \( \vec{F}(x, y) = \langle -y, x \rangle \). Then \[ \int_{C} \vec{F} \cdot d\vec{r} = \boxed{\phantom{0}} \] The problem involves evaluating a line integral over the top half of a circle centered at the origin with a radius of 4. The vector field \(\vec{F}(x, y)\) is defined as \(\langle -y, x \rangle\), which suggests a counterclockwise rotation of vectors around the origin. The goal is to calculate the integral of the dot product of the vector field and the differential arc length along the specified path \( C \).
Transcribed Image Text:Let \( C \) be the top half of the circle \( x^2 + y^2 = 16 \), and let \( \vec{F}(x, y) = \langle -y, x \rangle \). Then \[ \int_{C} \vec{F} \cdot d\vec{r} = \boxed{\phantom{0}} \] The problem involves evaluating a line integral over the top half of a circle centered at the origin with a radius of 4. The vector field \(\vec{F}(x, y)\) is defined as \(\langle -y, x \rangle\), which suggests a counterclockwise rotation of vectors around the origin. The goal is to calculate the integral of the dot product of the vector field and the differential arc length along the specified path \( C \).
Let \(\vec{G}(x, y) = \langle x, y \rangle\) and let \(C\) be parameterized by \(\vec{r}(t) = \langle 4 \cos(t), 4 \sin(t) \rangle\), \(0 \leq t \leq \pi\). Then \(\vec{G}(\vec{r}(t)) \cdot \vec{r}'(t) = \boxed{\phantom{aaa}}\) and \(\int_C \vec{G} \cdot d\vec{r} = \boxed{\phantom{aaa}}\)
Transcribed Image Text:Let \(\vec{G}(x, y) = \langle x, y \rangle\) and let \(C\) be parameterized by \(\vec{r}(t) = \langle 4 \cos(t), 4 \sin(t) \rangle\), \(0 \leq t \leq \pi\). Then \(\vec{G}(\vec{r}(t)) \cdot \vec{r}'(t) = \boxed{\phantom{aaa}}\) and \(\int_C \vec{G} \cdot d\vec{r} = \boxed{\phantom{aaa}}\)
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