I F. dr where C is oriented counterclockwise as viewed from above. F(x, y, z) = (x + y2)i + (y + z²)j + (z + x²)k, C is the triangle with vertices (4, 0, 0), (0, 4, 0), and (0, 0, 4). Step 1 Stokes' Theorem tells us that if C is the boundary curve of a surface S, then [ F. dr = £ £₂ curl F. ds. Since C is the triangle with vertices (4, 0, 0), (0, 4, 0), and (0, 0, 4), then we will take S to be the triangular region enclosed by C. The equation of the plane containing these three points is z = (-1)x+ (-1)y + ✓ Step 2 S is the portion of the plane z = 4 - x - y which lies over the region D in the xy-plane, where D is bounded by the x-axis, the y-axis, and the line y = (-1)x+ Step 3 For F(x, y, z) = (x + y²)i + (y + z²)j + (z + x²)k, we have curl F = (2=) i + -2x )j + (−2y )k. x Use Stokes' Theorem to evaluate
I F. dr where C is oriented counterclockwise as viewed from above. F(x, y, z) = (x + y2)i + (y + z²)j + (z + x²)k, C is the triangle with vertices (4, 0, 0), (0, 4, 0), and (0, 0, 4). Step 1 Stokes' Theorem tells us that if C is the boundary curve of a surface S, then [ F. dr = £ £₂ curl F. ds. Since C is the triangle with vertices (4, 0, 0), (0, 4, 0), and (0, 0, 4), then we will take S to be the triangular region enclosed by C. The equation of the plane containing these three points is z = (-1)x+ (-1)y + ✓ Step 2 S is the portion of the plane z = 4 - x - y which lies over the region D in the xy-plane, where D is bounded by the x-axis, the y-axis, and the line y = (-1)x+ Step 3 For F(x, y, z) = (x + y²)i + (y + z²)j + (z + x²)k, we have curl F = (2=) i + -2x )j + (−2y )k. x Use Stokes' Theorem to evaluate
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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![### Using Stokes' Theorem to Evaluate a Surface Integral
#### Problem Statement
Use Stokes' Theorem to evaluate the line integral \(\int_C \mathbf{F} \cdot d\mathbf{r}\) where \(C\) is oriented counterclockwise as viewed from above.
Given:
\[
\mathbf{F}(x, y, z) = (x + y^2)\mathbf{i} + (y + z^2)\mathbf{j} + (z + x^2)\mathbf{k}
\]
\(C\) is the triangle with vertices \((4, 0, 0)\), \((0, 4, 0)\), and \((0, 0, 4)\).
#### Step 1
Stokes' Theorem tells us that if \(C\) is the boundary curve of a surface \(S\), then:
\[
\int_C \mathbf{F} \cdot d\mathbf{r} = \iint_S \text{curl} \mathbf{F} \cdot d\mathbf{S}
\]
Since \(C\) is the triangle with vertices \((4, 0, 0)\), \((0, 4, 0)\), and \((0, 0, 4)\), then we will take \(S\) to be the triangular region enclosed by \(C\). The equation of the plane containing these three points is:
\[
z = (-1)x + (-1)y + 4
\]
#### Step 2
\(S\) is the portion of the plane \(z = 4 - x - y\) which lies over the region \(D\) in the \(xy\)-plane, where \(D\) is bounded by the \(x\)-axis, the \(y\)-axis, and the line:
\[
y = (-1)x + 4
\]
#### Step 3
For \(\mathbf{F}(x, y, z) = (x + y^2)\mathbf{i} + (y + z^2)\mathbf{j} + (z + x^2)\mathbf{k}\), we have:
\[
\text{curl} \mathbf{F} =
\left( \left( 2z \right) - \left( \right), \left( -2x](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F64590550-d676-4b57-9a24-39401c3be4ef%2F36870292-1e56-4335-89b0-fcde829a81a3%2F531p9u_processed.png&w=3840&q=75)
Transcribed Image Text:### Using Stokes' Theorem to Evaluate a Surface Integral
#### Problem Statement
Use Stokes' Theorem to evaluate the line integral \(\int_C \mathbf{F} \cdot d\mathbf{r}\) where \(C\) is oriented counterclockwise as viewed from above.
Given:
\[
\mathbf{F}(x, y, z) = (x + y^2)\mathbf{i} + (y + z^2)\mathbf{j} + (z + x^2)\mathbf{k}
\]
\(C\) is the triangle with vertices \((4, 0, 0)\), \((0, 4, 0)\), and \((0, 0, 4)\).
#### Step 1
Stokes' Theorem tells us that if \(C\) is the boundary curve of a surface \(S\), then:
\[
\int_C \mathbf{F} \cdot d\mathbf{r} = \iint_S \text{curl} \mathbf{F} \cdot d\mathbf{S}
\]
Since \(C\) is the triangle with vertices \((4, 0, 0)\), \((0, 4, 0)\), and \((0, 0, 4)\), then we will take \(S\) to be the triangular region enclosed by \(C\). The equation of the plane containing these three points is:
\[
z = (-1)x + (-1)y + 4
\]
#### Step 2
\(S\) is the portion of the plane \(z = 4 - x - y\) which lies over the region \(D\) in the \(xy\)-plane, where \(D\) is bounded by the \(x\)-axis, the \(y\)-axis, and the line:
\[
y = (-1)x + 4
\]
#### Step 3
For \(\mathbf{F}(x, y, z) = (x + y^2)\mathbf{i} + (y + z^2)\mathbf{j} + (z + x^2)\mathbf{k}\), we have:
\[
\text{curl} \mathbf{F} =
\left( \left( 2z \right) - \left( \right), \left( -2x
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