Let C be a triangle in the x-y plane with vertices (x1,y1), (x2y2) and (x3,y3) arranged so that C is positively-oriented.
Let C be a triangle in the x-y plane with vertices (x1,y1), (x2y2) and (x3,y3) arranged so that C...
Let C be a triangle in the ry-plane with vertices (ıv). (2.92), and (T3, Vs) arranged so that C' is positively-oriented a.) Sketch such a triangle and indicate its orientation. b.) Apply Green's Theorem to compute the area of the triangle as a (sum of) path integral(s) around the boundary. Get a formula for area in terms of the coordinates zi and y. This formula is sometimes referred to as the Shoelace formula for area. c.) Use the formula you...
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...
Let F(x, y,z) = < x + y2,y + z2,z + x2 >, let S be a surface with boundary C. C is the triangle with vertices (1,0,0), (0,1,0), (0,0,1). 8. a. Evaluate F dr curl F ds b. Let F(x, y,z) = , let S be a surface with boundary C. C is the triangle with vertices (1,0,0), (0,1,0), (0,0,1). 8. a. Evaluate F dr curl F ds b.
4. Let - xy’i +3yj , and let C be the counterclockwise oriented triangle whose vertices are 0(0,0), P(2,0) and Q(2,8). Using Green's Theorem, ايثار a) Sf(-2x)dyex b) ſj(-2xy)dydx c) ff(xv* + 3y Mdvdx 0555(xv? +3 y Ddydx e) none of these 00 0 0
(a) Let C be the line segment on the plane that starts from a point (xi,yi) to a different point (x2,Y2). Show that (b) Consider a simple polygon whose vertices are (2.1 , Й), (T2, Уг), . . . , (Xn, yn) if its boundary is traversed counterclockwise. Use Green's theorem to show that the area of this polygon is (a) Let C be the line segment on the plane that starts from a point (xi,yi) to a different point...
Let A be the triangle in the two-dimensional plane with vertices (0, 0), (0, 1), and (1, 0). Let (X, Y ) be chosen uniformly from this area, that is, (X, Y ) ∼ Unif(A). (a) What is the probability that X ≤ 1/3? (b) What is the probability that Y ≥ 1/2? (c) Conditioned on X ≤ 1/3, what is the probability that Y ≥ 1/2? (d) Are the events X ≤ 1/3 and Y ≥ 1/2 independent?
(23 pts) Let F(x, y, z) = ?x + y, x + y, x2 + y2?, S be the top hemisphere of the unit sphere oriented upward, and C the unit circle in the xy-plane with positive orientation. (a) Compute div(F) and curl(F). (b) Is F conservative? Briefly explain. (c) Use Stokes’ Theorem to compute ? F · dr by converting it to a surface integral. (The integral is easy if C you set it up correctly) 4. (23 pts)...
All of 10 questions, please. 1. Find and classify all the critical points of the function. f(x,y) - x2(y - 2) - y2 » 2. Evaluate the integral. 3. Determine the volume of the solid that is inside the cylinder x2 + y2- 16 below z-2x2 + 2y2 and above the xy - plane. 4. Determine the surface area of the portion of 2x + 3y + 6z - 9 that is in the 1st octant. » 5. Evaluate JSxz...
7. Use Green's Theorem to find Jc F.nds, where C is the boundary of the region bounded by y = 4-x2 and y = 0, oriented counter-clockwise and F(x,y) = (y,-3z). what about if F(r, y) (2,3)? x2 + y2 that lies inside x2 + y2-1. Find the surface area of this 8. Consider the part of z surface. 9. Use Green's Theorem to find Find J F Tds, where F(x, y) (ry,e"), and C consists of the line segment...
Determine whether the following are true or false: A) If Sis a surface parametrized byr:DR^3, then A(S) = (double integral)D dA, where A(S) is the surface area of S. B) Let c be a boundary of a closed and bounded region D in the xy-plane. Then counterclockwise is always a positive orientation of c. c) Let Fbe a constant vector field on R^3. Then the flux of F through the unit sphere x^2 + y^2 + 2^2 = 1 is...