Proving the Fundamental Theorem for Line Integrals Let F be the vector field F = Mi + Nj + Pk, so...
#3 Consider the vector field F- Mi+ Nj Pk defined by: F- ysinzi+sinjry cos z k. Compute the line integral ScF dr over a unit circle. Compute the line integral ysin z dr+ r sin z dy + ry cos zdz (0,0,0) #3 Use Green's Theorem to evaluate the line integral along the given positively orientated curve C. e2*t d e" dy, C is the triangle with vertices (0,0), (1,0), and (1,1) #3 Consider the vector field F- Mi+ Nj...
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).
(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
Let F(x, y, z) = sin yi + (x cos y + cos z)j – ysin zk be a vector field in R3. (a) Verify that F is a conservative vector field. (b) Find a potential function f such that F = Vf. (C) Use the fundamental theorem of line integrals to evaluate ScF. dr along the curve C: r(t) = sin ti + tj + 2tk, 0 < t < A/2.
10. Stokes' Theorem and Surface Integrals of Vector Fields a. Stokes' Theorem: F dr- b. Let S be the surface of the paraboloid z 4-x2-y2 and C is the trace of S in the xy-plane. Draw a sketch of curve C in the xy-plane. Let F(x,y,z) = <2z, x, y?». Compute the curl (F) c. d. Find a parametrization of the surface S: G(u,v)- Compute N(u,v) e. Use Stokes' Theorem to computec F dr 10. Stokes' Theorem and Surface Integrals...
10. Stokes' Theorem and Surfac e Integrals of Vector Fields a. Stokes' Theorem: F-dr= b. Let S be th ky-plane. Draw a sketch of curve C in the xy-plane. et be the surface of the paraboloid z 4-x-y and Cis the trace of S in the c Let Fox.y.z) <2z, x, y>, Compute the curl (F) d. Find a parametrization of the surface S: G(u,v)- Compute N(u,v) F-dr Use Stokes' Theorem to compute , e. 10. Stokes' Theorem and Surfac...
10. Stokes Theorem and Surface Integrals of Vector Fields a Stokes Theorem:J F dr- b. Let S be the surface of the paraboloid z 4-x2-y2 and C is the trace of S in the xy-plane. Draw a sketch of curve C in the xy-plane. Let F(x,y,z) = <2z, x, y, Compute the curl (F) c. d. Find a parametrization of the surface S: G(u,v)ーーーーーーーーーーーーー Compute N(u,v) e. Use Stokes' Theorem to compute Jc F dr 10. Stokes Theorem and Surface...
b- Consider the vector field F(x,y,z)= (3x²y2-3ze, 2xy +2sin z, - 3x02 + 2ycos z). (a) if f(x,y,z) = axºy2+be*2 + cysin z then a =......, (b) Use the fundamental theorem of LINE INTEGRAL to evaluate Y = SF-di along the curve defined by the parametrization F(t)= (1, sint, t-T) for Osts. Y = ...... b Choose... Y = Choose.... Choose... Choose...
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...
Let F = (2,1,1) be a vector field in Rº a.) Show that F is a conservative vector field. b.) Find the potential function of F. In other words, find a scalar function f(x, y, z) such that ✓ f = 7. Please show all steps. c.) Let C be any smooth curve starting at (1,1,1) and ending at (e, e, 1). Compute (Fdi. С