Find the vector field F such that if Green’s Theorem is applied to F and the region bounded by C is D = {(x, y) : a ≤ x ≤ b, 0 ≤ y ≤ c}, then the result is the fundamental of calculus from Calc 1. Justify why the desired result holds.
Find the vector field F such that if Green’s Theorem is applied to F and the region bounded by C ...
3) (11 points) Consider the vector field Use the Fundamental Theorem of lLine Integrals to find the work done by F along any curve from 41. 1Le) to B(2. el) 4) (10 points) Consider the vector field F(x.y)-(r-yi+r+y)j and the circle C: r y-9. Verify Green's Theorem by calculating the outward flux of F across C (12 points) Find the absolute extreme values of the function .-2-4--3 on the closed triangular region in the xy-plane bounded by the lines x...
calc 3 7) Fundamental Theorem of Line Integrals. a) Show that the vector field, F(x,y) = (2x - 2)i - 23e2v j, is conservative. b) Find a potential function for F. c) Evaluate F. dr if C is the path connecting the three line segments from (1,0) to (2,5) then from (2,5) to (-2,5) and finally from (-2,5) to (-1,0).
1. [-/5 Points) DETAILS 0/1 Submissions Used Find the area of the region bounded by the graphs of the equations. y = 6 + x x = 0, X = 8, y = 0 2. (-/5 Points] DETAILS 0/1 Submissions Used Use the Second Fundamental Theorem of Calculus to find F"(x). F(x) = - 6". 8t cos(t) dt F'(x) =
4. Consider the vector field u = (3r+yz) region V bounded by 2y2 < (2 - z)2 for y 2 0 and 0 y)j+(xy+2z)k, defined across a three-dimensional 1. z (a) Show that u is conservative and find a scalar function d that satisfies u = Vo. [6 marks] (b) Sketch the volume V and express the limits of the volume V in terms of cylindrical coordi nates (r, 0, z) [3 marks (c) Using the divergence theorem calculate the...
Proving the Fundamental Theorem for Line Integrals Let F be the vector field F = Mi + Nj + Pk, so 1. Assurne F is a gradient vector field with potential function f(x, y, z). Let x = x(t), y = y(t),z(t), a < t S b be a parametrization of the curve C, starting at P, ending at Q Explain why this means Proving the Fundamental Theorem for Line Integrals Let F be the vector field F = Mi...
3. (10 pts) Find the area of the region bounded between y = xe-*?, , y = x + 1, x = 2 and the y-axis. Note that the graph of the region is provided below. You can leave your answer in terms of e. y=x+1 x2 X-0 0 0.5 1. 0 dy Use the Fundamental Theorem of Calculus to find dx for y = = L* sin (t2)dt.
(1 point) Verify the Divergence Theorem for the vector field and region: F-(2x, 82.9y〉 and the region x2 + y2-1, 0-X 7 (1 point) Verify the Divergence Theorem for the vector field and region: F-(2x, 82.9y〉 and the region x2 + y2-1, 0-X 7
consider a simple smooth closed curve C and a vector field F= Mi+Nj verifying the conditions of both forms of green’s theorem. Find a vector G=Pi+Qj (that is write P and Q in function of M and N) such that the counterclockwise circulation of F along C = the outward flux of G across C.
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 +...
Use the divergence theorem to find the outward flux(F n) dS of the given vector field F y21+ xz3j + (z-1)2k; D the region bounded by the cylinder x2 + y2-36 and the planes z-1,2 F 1, Z 9 eBook