3. Consider A(r) = (y, –x, 0), and the line integral ØA• dr along theCboundary of the plane x + y + z = 1 in the first octant, oriented asshown in Figure 2.Convert the line integral into a surface integral through Stokes'Theorem, then evaluate the surface integral.Figure 1Figure 2

The given vector function is A(r)=(y,-x,0)., the line integral ∮CA·dr is along the boundary of the…

Answered: 3. Consider A(r) = (y, -x,0), and the…

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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Consider the surface z(x, y) = x²y = 3 sn (x√y), find the equation of the tangent plane to the surface point P(pi/3, 1). produce answer in exact form.
4. Let S is the surface z = 5 - r - y, 1<: in R. Let S be oriented with the normal vector pointing upward direction. (a) Let F(r, y, 2) = (y., z, 3). Use Stokes Theorem to evaluate (b) Verify Stokes' theorem for part (a).
Use Stokes's Theorem to show that 0 = |J curl(F) - dS where F(r, y, z) = (z, 2ry, x + y) and S is the surface of a glass oriented outwards whose open cap is a circle with equation r² + (z – 2)² = 9.
Let S be given by x² + y² + z² = 25 and z > 0, oriented downward. (a) Sketch S and aS. Indicate their orientations on your sketch. Parametrize aS with the correct orientation. (b) Let F(x, y, z) = (z², x, y²). Verify Stokes' theorem.
Let F = Use Stokes' Theorem to evaluate So F. dr, where C is the curve of intersection of the parabolic cylinder z = y² - x and the circular cylinder x² + y² = 1, oriented counterclockwise as viewed from above.
How to solve this question
6. Use the Stokes theorem to evaluate f Fdr where F (3x – 2y +32) i +(4x – 2y+32) 5 + (2x – 3y | 22) k and C is the circle: æ = 3, y? | 22 = 4, oricnted counterclockwisc when vicwcd from the positive part of x-axis. Describe the surface S whose boundary is C and state its orientation.
Use Stokes' Theorem to find the circulation of F = (ry, yz, rz) around the boundary of the surface S given by z = 9 -a for 0

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3. Consider A(r) = (y, -x,0), and the line integral A• dr along the C boundary of the plane x + y + z = 1 in the first octant, oriented as shown in Figure 2. Convert the line integral into a surface integral through Stokes' Theorem, then evaluate the surface integral.