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E Let C₁ be the curve parameterized by r(t) = (cos² t, sint) where t = [-]. Let C₂ be the line segment from (0, 1) to (0, -1). (a) Points (x, y) on curve C₁ all lie on the graph of an equation x = f(y) for some simple function f. What is that function? (b) Use Green's theorem to find the flux of F = xy² i + (y- ) j out of the region bounded between C₁ and C₂. (c) Use your answer to part (b) to find the left-to-right flux of F = xy² i+(y-ez³)j across C₁ only without having to take the flux integral over C₁ directly.

E Let C₁ be the curve parameterized by r(t) = (cos² t, sint) where t = [-]. Let C₂ be the line segment from (0, 1) to (0, -1). (a) Points (x, y) on curve C₁ all lie on the graph of an equation x = f(y) for some simple function f. What is that function? (b) Use Green's theorem to find the flux of F = xy² i + (y- ) j out of the region bounded between C₁ and C₂. (c) Use your answer to part (b) to find the left-to-right flux of F = xy² i+(y-ez³)j across C₁ only without having to take the flux integral over C₁ directly.

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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**Problem 2: Vector Calculus and Green’s Theorem** Consider the following curves and tasks: **Curve Definitions:** - Let \( C_1 \) be the curve parameterized by \(\mathbf{r}(t) = (\cos^2 t, \sin t)\) where \( t \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \). - Let \( C_2 \) be the line segment from \((0, 1)\) to \((0, -1)\). **Tasks:** **(a)** Determine the function: - Points \((x, y)\) on the curve \( C_1 \) all lie on the graph of an equation \( x = f(y) \) for some simple function \( f \). Identify this function. **(b)** Apply Green’s Theorem: - Use Green’s theorem to find the flux of the vector field \(\mathbf{F} = xy^2 \mathbf{i} + (y - e^{x^3}) \mathbf{j} \) out of the region bounded between \( C_1 \) and \( C_2 \). **(c)** Calculate Flux for Curve \( C_1 \): - Use your result from part (b) to find the left-to-right flux of \(\mathbf{F} = xy^2 \mathbf{i} + (y - e^{x^3}) \mathbf{j} \) across \( C_1 \) only, without directly computing the flux integral over \( C_1 \). Use mathematical principles and calculations to aid in determining solutions for each part.
Transcribed Image Text:**Problem 2: Vector Calculus and Green’s Theorem** Consider the following curves and tasks: **Curve Definitions:** - Let \( C_1 \) be the curve parameterized by \(\mathbf{r}(t) = (\cos^2 t, \sin t)\) where \( t \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \). - Let \( C_2 \) be the line segment from \((0, 1)\) to \((0, -1)\). **Tasks:** **(a)** Determine the function: - Points \((x, y)\) on the curve \( C_1 \) all lie on the graph of an equation \( x = f(y) \) for some simple function \( f \). Identify this function. **(b)** Apply Green’s Theorem: - Use Green’s theorem to find the flux of the vector field \(\mathbf{F} = xy^2 \mathbf{i} + (y - e^{x^3}) \mathbf{j} \) out of the region bounded between \( C_1 \) and \( C_2 \). **(c)** Calculate Flux for Curve \( C_1 \): - Use your result from part (b) to find the left-to-right flux of \(\mathbf{F} = xy^2 \mathbf{i} + (y - e^{x^3}) \mathbf{j} \) across \( C_1 \) only, without directly computing the flux integral over \( C_1 \). Use mathematical principles and calculations to aid in determining solutions for each part.
Expert Solution
Step 1: Writing the function satisfying the parameterized curve

Given that C1 is the curve parameterized by r(t)=(cos2⁡t,sin⁡t),t∈[−π2,π2].

Let C2 be the line segment from (0,1) to (0,−1).

(a) The points (x,y) lie on C1.

Let x equals cos squared t and y=sin⁡(t)

then

table row cell cos squared t end cell equals cell 1 minus sin squared t end cell row x equals cell 1 minus y squared end cell end table

Thus, the function is f(y)=1−y2.

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