15(20 points) Find the value of the integral Cis a counterclockwise oriented circle. C: 13-j-2
(a) (15 F-(1+9) 9. points) Apply Green's theorem to evaluate φ F.nds, where (x2 +y)j, of a triangle with vertices (1,0), (0,1). (-1,0) oriented in the counterclockwise direction n is the outward-pointing normal vector on , and C is the boundary (b) (15 points) Evaluate directly the line integral p F- nds in part (a).
(a) (15 F-(1+9) 9. points) Apply Green's theorem to evaluate φ F.nds, where (x2 +y)j, of a triangle with vertices (1,0), (0,1). (-1,0) oriented in...
Evaluate the integral
3,3 + 2 where C is the positively oriented circle 2-22
6. (1 point) Use Stokes' Theorem to find the line integral /2y dx + dy + (4-3x) dz, where C is the boundary of the triangle with vertices (0,0,0), (1,3,-2), and -2,4,5), oriented counterclockwise as viewed from the point (1, 0, 0)
6. (1 point) Use Stokes' Theorem to find the line integral /2y dx + dy + (4-3x) dz, where C is the boundary of the triangle with vertices (0,0,0), (1,3,-2), and -2,4,5), oriented counterclockwise as viewed from the...
10. Use Green's theorem to find f dr where 1 F(x,y) -2,23, อี่+ry2 and C is the circle 2,2 +Y'2 4 oriented counterclockwise.
10. Use Green's theorem to find f dr where 1 F(x,y) -2,23, อี่+ry2 and C is the circle 2,2 +Y'2 4 oriented counterclockwise.
(20 points) Let and let C' be any simple closed curve in a plane oriented counterclockwise. Please show that the only two possible values for F. dr is 0 or-2π. (Hint) The domain of the vector field does not include the origin. Hence, the origin is seen as a hole. Consider 1) Curve C does not encompass the origin. 2) Curve C does encompass the origin. In this case, use an auxiliary curve that encompasses the origin and is encompassed...
Use Green's Theorem to evaluate the integral. Assume that the curve C is oriented counterclockwise. ∮C 6 ln(6+y) dx−(xy/6+y) dy, where C is the triangle with vertices (0,0), (6,0), and (0,12) ∮C 6 ln(6+y) dx−(xy/6+y)dy=
Use Green's Theorem to evaluate the integral. Assume that the curve C is oriented counterclockwise. ху 7 In(7 + y) dx - dy, where C is the triangle with vertices (0,0), (4,0), and (0,8) fe 7+ y ху f 7 ln(7 + y) dx – dy = 7+y
9. [15 Points) Let C be the boundary of the triangle with vertices (1, 1), (2, 3) and (2, 1), oriented positively i.e. counterclockwise). Let F be the vector field F(1, y) = (e* + y²)i + (ry + cos y)j. Compute the line integral F. dr. 10. (15 Points) Let S be the portion of the paraboloid z = 1-rº-ythat lies on and above the plane z = 0. S is oriented by the normal directed upwards. If F...
(5 points.) Let C be the positively oriented circle of radius 2 around the origin. The mapping w 1/(2(22-1(22-9)) transforms C into a closed curve I. Find the winding number of 1.
(5 points.) Let C be the positively oriented circle of radius 2 around the origin. The mapping w 1/(2(22-1(22-9)) transforms C into a closed curve I. Find the winding number of 1.
Use Stokes' Theorem to evaluate F. dr where Cis oriented counterclockwise as viewed from above. F(x, y, z) - xy + 27 + 6yk, C is the curve of intersection of the plane X + 2-1 and the cylinder + 9.