3. Let Hi, y) = (2xy)i + (x2 +2y) Then for any piecewise smooth oriented عاريا...
Let F(x, y) = (2xy)i + (x?+ 2y). Then for any piecewise smooth oriented path C beginning at (0,0) and ending at (1,3), SF•di = a) 3b) 6 c) 9 d) 12 e) none of these
5. Let F (y”, 2xy + €35, 3yes-). Find the curl V F. Is the vector field F conservative? If so, find a potential function, and use the Fundamental Theorem of Line Integrals (FTLI) to evaluate the vector line integral ScF. dr along any path from (0,0,0) to (1,1,1). 6. Compute the Curl x F = Q. - P, of the vector field F = (x4, xy), and use Green's theorem to evaluate the circulation (flow, work) $ex* dx +...
(6) Fundamental Theorem of Line Integrals F = <M,N> = < 2xy, x² + y2 > (6a) Show that F is a Conservative Vector Field. (6b) Find the Potential Function f(x,y) for the Vector Field F. (60) Evaluate W = | Mdx + Ndy from (5,0) to (0,4) over the path C: È + K3 = 1 с
Use to fundamental theorem of line integrals to evaluate F dr for 6. F(xy) = (2xy,x2 -y) over the path C from the point (2, 0) to (0, 2) Use to fundamental theorem of line integrals to evaluate F dr for 6. F(xy) = (2xy,x2 -y) over the path C from the point (2, 0) to (0, 2)
3. Consider the vector field F(x, y) + 2y F dr, where C is the circle (r-2)2 +y2 = 1, oriented counterclock (a) Compute wise (Hint: use the FT of line integrals. We could not use it for the circle centered at the origin, but we can use the theorem for this circle. Why?) (b) Let 0 be the angle in polar coordinates for a point (x, y). Check that 0 is a potential function for F 3. Consider the...
6. Calculate the following line integrals of vector fields. Be sure to name any theorems you use; if you don't use a theorem, write "calculated directly2 (d) F . dr, where F(x,y)-(2ry-уг, r2 +3y2-2cy), and C is the piecewise-linear path frorn (1,3) to (5,2 to (12) to (4,1) (e) φ F.dr, where F(z,y)-(3ysin(Zy), 3rsin(2y)+6ry cos(2p)), and C is the ellipse 2 +9y2-64. oriented counter-clockwise 6. Calculate the following line integrals of vector fields. Be sure to name any theorems you...
If we start with o and form F from it, we are definitely creating a co Let's start there. 4. Suppose that Q(x, y?). Let F(x,y) = Vo(x,y). a. Find Vé(x,y). F.Tds if C is the quarter unit circle from (1,0) to (0,1). b. Let F(x,y)=VQ(x,y). Find otomo 19 Il Fundamental Theorem for Line Integrals Let F be a continuous vector field on an open region R in R. There exists a potential function o with F= Vo (which means...
7. Assume (x, y,x)(2xy, y',5z - y). Let E be the solid upright cylinder between the planes z 0 and z-3 with base the disc x2 + y2 < 9, and let S be the outwardly oriented boundary surface of E. Note that S consists of three smooth surfaces; the surface Si of the cylinder, plus the top disc Di and the bottom disc D2. Follow the steps to verify the Divergence Theorem. (a) [12 pts.] Evaluate dS directly 7....
Please show all the work to complete the question and explain each step, please. Thank you! Let F(x, y) e*y (y cos x - centered at (1,0) in the first quadrant, traced clockwise from (0,0) to (2, 0). And suppose that C2 is the line from (0,0) to (2,0). sin x) xexy cos xj. Suppose that C1 is the half of the unit circle (A) Use the curl test to determine whether F is a gradient vector field or not....
QB(27pts)(a). Evaluate the circulation ofF(xy)-<x,y+x> on the curve r(t)=<2cost, 2sinp, foross2n (b) Evaluate J F.dr, where C is a piecewise smooth path from (1,0) to (2,1) and F- (e'cos x)i +(e'sinx)j [Hint: Test F for conservative (c). Use green theorem to express the line integral as a double integral and then evaluate. where C is the circle x+y-4 with counterclockwise orientation. (d(Bonus10 pts) Consider the vector field Foxyz) a. Find curl F y, ,z> F.dr where C is the curve...