Let C be a triangle in the ry-plane with vertices (ıv). (2.92), and (T3, Vs) arranged so that C' ...
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 xy-plane with vertices (x,y), (z2,p), and (z3,U3) 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...
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...
T3 finite element is defined over AABC (in physical coordinates). The vertices of this triangle hav the following coordinates: A(-2,-1), B(3,2), and C(0,6). Problem 2 a) Using 1 point and 3 point integration rules, compute f(x, y)ds AABC 2x2-3xy + y2. where f(x,y) Which rule gives more accurate result? c) What is the integration error, if 3 point rule is used? (Hint: for what polynomial degree 3 point rule gives the exact result?) b) T3 finite element is defined over...
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
Let A be the inside and boundary of the triangle in R2 whose vertices are (0,0), (1,0) and (0,1). Let C be the curve obtained by proceeding around the boundary of A in an anti- clockwise direction. Prove İ}!").lx (ly İ)(2 dr dy. Pdr+Qdy That is, prove Green's Theorem for the triangle A. [Hint: the lecture notes have a proof for when A is a rectangle. So, the idea is is to give a similar proof where we have this...
7. Evaluate (6x - 6y+8) dx+(4 +9y +7) dy where C is the boundary of the triangle in the ry plane, wit h vertices at (0,0), (1,0)and (1,4) traversed once anticlockwise. (a) 10 (c) 20 (b)-8 (d) 8 10. Find the flux of F =-rit 2yj otward across the ellipse-+ -1. (a) 36π (b) 18m (c) o (d) 6π 7. Evaluate (6x - 6y+8) dx+(4 +9y +7) dy where C is the boundary of the triangle in the ry plane,...
(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...
need 1-5 Midterm #3, Math 228 Each question is worth five points. 1. Let F(r.yzy). Let C be any curve that goes from A(-1,3,9) to B(1,6,-4). a) Show that F is conservative. b) Find a function φ such that ▽φ = F c) Use the result of b) to find Ic F Tds 2. Let F(z, y)-(2), and let C be the boundary of the square with vertices (1, 1). (-1,1). (-1,-1 traced out in the counter-clockwise direction. Find Jc...
3. (2 Points) Let Q be the quadrilateral in the ry-plane with vertices (1, 0), (4,0), (0, 1), (0,4). Consider 1 dA I+y Deda (a) Evaluate the integral using the normal ry-coordinates. (b) Consider the change of coordinates r = u-uv and y= uv. What is the image of Q under this change of coordinates?bi (c) Calculate the integral using the change of coordinates from the previous part. Change of Variables When working integrals, it is wise to choose a...