Question

Let F = P i + Q j = (x + y) i - (x^2 + y^2) j and consider the triangle with vertices (1, 0), (0, 1), and (-1, 0). You need to compute the outward flux of F across the boundary, C, of the triangle. Recalling that n = T × k, the flux is given by ?_C F · n ds = ?_C P dy - Q dx (a) Make a sketch of the region, indicating the proper orientation of the boundary of the triangle, C, for subsequent use of Green's Theorem. (b) Is the boundary of the triangle smooth? piecewise smooth? (c) How many line integrals will be required to compute the flux? (d) Let C_1 be the line segment from (1, 0) to (0, 1). Evaluate ?_C_1 P dy - Q dx. (e) To gain more familiarity with the notation, evaluate ?_C_2 F · n ds, where C_2 is the line segment from (0, 1) to (-1, 0). (f) By inspection, what is n on the final boundary, C_3, the line segment from (-1, 0) to (1, 0)? (g) Evaluate ?_C_3 P dy - Q dx, where C_3 is as in part (f). (h) Finally, what is the outward flux of F through C? (i) That was a lot of work. There is an easier way! The flux form of Green's Theorem can come to the rescue to lessen the workload. This is ?_C F · n ds = ?_C P dy - Q dx = ?_D (?P/?x + ?Q/?y) dA where D is the triangle. Use this theorem to compute the outward flux of F across C.

          Let F = P i + Q j = (x + y) i - (x^2 + y^2) j and consider the triangle with vertices (1, 0), (0, 1), and (-1, 0). You need to compute the outward flux of F across the boundary, C, of the triangle. Recalling that n = T × k, the flux is given by
?_C F · n ds = ?_C P dy - Q dx
(a) Make a sketch of the region, indicating the proper orientation of the boundary of the triangle, C, for subsequent use of Green's Theorem.
(b) Is the boundary of the triangle smooth? piecewise smooth?
(c) How many line integrals will be required to compute the flux?
(d) Let C_1 be the line segment from (1, 0) to (0, 1). Evaluate ?_C_1 P dy - Q dx.
(e) To gain more familiarity with the notation, evaluate ?_C_2 F · n ds, where C_2 is the line segment from (0, 1) to (-1, 0).
(f) By inspection, what is n on the final boundary, C_3, the line segment from (-1, 0) to (1, 0)?
(g) Evaluate ?_C_3 P dy - Q dx, where C_3 is as in part (f).
(h) Finally, what is the outward flux of F through C?
(i) That was a lot of work. There is an easier way! The flux form of Green's Theorem can come to the rescue to lessen the workload. This is
?_C F · n ds = ?_C P dy - Q dx = ?_D (?P/?x + ?Q/?y) dA
where D is the triangle. Use this theorem to compute the outward flux of F across C.

Let F = P i + Q j = (x + y) i - (x^2 + y^2) j and consider the triangle with vertices (1, 0), (0, 1), and (-1, 0). You need to compute the outward flux of F across the boundary, C, of the triangle. Recalling that n = T × k, the flux is given by
?C F · n ds = ?C P dy - Q dx
(a) Make a sketch of the region, indicating the proper orientation of the boundary of the triangle, C, for subsequent use of Green's Theorem.
(b) Is the boundary of the triangle smooth? piecewise smooth?
(c) How many line integrals will be required to compute the flux?
(d) Let C1 be the line segment from (1, 0) to (0, 1). Evaluate ?C1 P dy - Q dx.
(e) To gain more familiarity with the notation, evaluate ?C2 F · n ds, where C2 is the line segment from (0, 1) to (-1, 0).
(f) By inspection, what is n on the final boundary, C3, the line segment from (-1, 0) to (1, 0)?
(g) Evaluate ?C3 P dy - Q dx, where C3 is as in part (f).
(h) Finally, what is the outward flux of F through C?
(i) That was a lot of work. There is an easier way! The flux form of Green's Theorem can come to the rescue to lessen the workload. This is
?C F · n ds = ?C P dy - Q dx = ?D (?P/?x + ?Q/?y) dA
where D is the triangle. Use this theorem to compute the outward flux of F across C.

Added by Jasmine K.

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