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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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Question
![Suppose \(\vec{F}(x, y) = -y \hat{\imath} + x \hat{\jmath}\) and \(C\) is the line segment from point \(P = (3, 0)\) to \(Q = (0, 4)\).
(a) Find a vector parametric equation \(\vec{r}(t)\) for the line segment \(C\) so that points \(P\) and \(Q\) correspond to \(t = 0\) and \(t = 1\), respectively.
\[
\vec{r}(t) = \boxed{ }
\]
(b) Using the parametrization in part (a), the line integral of \(\vec{F}\) along \(C\) is
\[
\int_C \vec{F} \cdot d\vec{r} = \int_a^b \vec{F}(\vec{r}(t)) \cdot \vec{r}'(t)\, dt = \int_a^b \boxed{ } \, dt
\]
with limits of integration \(a = \boxed{ }\) and \(b = \boxed{ }\).
(c) Evaluate the line integral in part (b).
\[
\boxed{ }
\]
(d) What is the line integral of \(\vec{F}\) around the clockwise-oriented triangle with corners at the origin, \(P\), and \(Q\)? Hint: Sketch the vector field and the triangle.
\[
\boxed{ }
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7e0c8037-834b-4d36-b3c4-eda43340784b%2Fc5180fb0-4f2e-46dd-a8f1-eff925a44cf5%2F7l5rsr_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Suppose \(\vec{F}(x, y) = -y \hat{\imath} + x \hat{\jmath}\) and \(C\) is the line segment from point \(P = (3, 0)\) to \(Q = (0, 4)\).
(a) Find a vector parametric equation \(\vec{r}(t)\) for the line segment \(C\) so that points \(P\) and \(Q\) correspond to \(t = 0\) and \(t = 1\), respectively.
\[
\vec{r}(t) = \boxed{ }
\]
(b) Using the parametrization in part (a), the line integral of \(\vec{F}\) along \(C\) is
\[
\int_C \vec{F} \cdot d\vec{r} = \int_a^b \vec{F}(\vec{r}(t)) \cdot \vec{r}'(t)\, dt = \int_a^b \boxed{ } \, dt
\]
with limits of integration \(a = \boxed{ }\) and \(b = \boxed{ }\).
(c) Evaluate the line integral in part (b).
\[
\boxed{ }
\]
(d) What is the line integral of \(\vec{F}\) around the clockwise-oriented triangle with corners at the origin, \(P\), and \(Q\)? Hint: Sketch the vector field and the triangle.
\[
\boxed{ }
\]
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