Question 2. In class we proved that if proved that if /F. dr is independent of...
tion 2. In class we proved that if proved that if ſc F. dr is independent of path in a domain D, then F F. dr = 0 for any closed curve C CD. Prove the converse: ie, prove that if Sc F. dr = 0 for any closed curve C C D, then Sc F. dr is independent of path in D. Hint: Choose two points A and B in D, and form two paths C1 and C2 from...
(1 point) Are the following statements true or false? (1 point) Are the following statements true or false? v 1. If the vector fields F and Ğ Have ScĘ. dr = ScĞ. dr for a particular path C, then F=G. 2. The circulation of any vector field Ę around any closed curve C is zero. ? ? ? 3. If F = Vf, then F is conservative. 2 4. If ScĚ.dñ = 0 for one particular closed path, then F...
please give some explanation to each step 15 Total Question 3 Let F: R3R3 be any C2 vector field. 3(a). Prove that the divergence of the curl of F is zero. /4 marks 3(b). For F as defined above, a misguided professor claims that for any closed curve C, F dr 0 because of the argument: (x F)ds F-dr div (eurl F) dV X 0-APO by using Stokes' theorem, the divergence theorem, and then part (a) for an appropriately chosen...
3. Evaluate |F.dr w tine integral is independent of path. Compute F-dr and C is the ellipse given with the counter clockwise rotation Answer by 7-6--4 4 Evaluate vf-dr where (x.y)-d C is the curve shown below. Answer. |vf.dr=-4 3. Evaluate |F.dr w tine integral is independent of path. Compute F-dr and C is the ellipse given with the counter clockwise rotation Answer by 7-6--4 4 Evaluate vf-dr where (x.y)-d C is the curve shown below. Answer. |vf.dr=-4
1. One of the two vector fields listed below is conservative. The other one is not conservative. (a) Determine which one of these fields is conservative. Label the conservative field F and and find a potential function f for it. Label the other field G and prove that G is NOT conservative. (b) Use the fundamental theorem of line integrals to compute SCF . dr, where C is the curve parameterized by (c) Compute Jc G-dr, where C is the...
(2 pts) Calculate the circulation, rF dr, in two ways, directly and using Stokes' Theorem. The vector field F (8x-8y+62)(i + j) and C is the triangle with vertices (0,0,0), (8, 0, 0), (8,2,0), traversed in that order. Calculating directly, we break C into three paths. For each, give a parameterization r (t) that traverses the path from start to end for 0sts 1 On Ci from (0,0, 0) to (8,0,0), r(t) = <8t,0,0> On C2 from (8, 0, 0)...
True or False Determine whet her the statement is true or false, and circle the correct answer. Each question is worth 2 points. (1) If F is a vector field and C is an oriented curve, then F dr must be less than zero. F (2) It is possible that for a certain vector field F and piecewise smooth oriented path C we have/. F. dr-2i-Sj. (3) Suppose d·is the unit square joining the points (0,0), (1,0), (1,1), (0.1) oriented...
2. (a) Let i. Show that F is cnservative in R i. Let C denote the path 1+cost,2+sint,3), 0StS 4 Evaluate F. dr Justify your answer. iii. Find a function y: R3-+ R such that F iv. Evaluate F.dr where「is the path y =r', z = 0, from (0.0.0) to (2.8.0) followed by the line segment from (2,8,0) to (1,1,2) 22 marks) 2. (a) Let i. Show that F is cnservative in R i. Let C denote the path 1+cost,2+sint,3),...
9. (10pts) Answer true or false: (a) The domain of f(x,y) = In(1-z?-уг) + Vi-z?-уг is the unit ball {(z, y): x2 + y2 1} . (b) The direction of the maximum rate of increase of g(x, y, 2yz at the point (1,1,1) is 2,1,1 (c) For F2y,2r3y1>, F-dr is independent of path in the plane. (d) × (▽ . F) makes sense. (e) ▽f.dr =4 where f(x, y, z) = zyz and C is the line segment starting at...
An important fact we have proved is that the family (enr)nez is orthonormal in L (T,C) and complete, in the sense that the Fourier series of f converges to f in the L2-norm. In this exercise, we consider another family possessing these same properties. On [-1, 1], define dn Ln)-1) 0, 1,2, Then Lv is a polynomial of degree n which is called the n-th Legendre polynomial. (a) Show that if f is indefinitely differentiable on [-1,1], thern In particular,...