Suppose F⃗(x,y)=(x+6)i⃗+(5y+5)j⃗. Use the fundamental theorem of line integrals to calculate the following
(a) The line integral of F⃗→ along the line segment C from the
point P=(1,0) to the point Q=(4,2).
∫CF⃗⋅dr⃗∫=
(b) The line integral of F⃗→ along the triangle C from the
origin to the point P=(1,0) to the point Q=(4,2) and back to the
origin.
∫CF⃗⋅dr⃗∫=
Suppose F⃗(x,y)=(x+6)i⃗+(5y+5)j⃗. Use the fundamental theorem of line integrals to calculate the following (a) The line integral of F⃗→ along the line segment C from the point P=(1,0) to the point Q=(...
Suppose that is the path which starts at the point (2,0), then moves along the circle4 to the point (0,2). Next, the path moves along the line segment from (0,2) to (4,3). Calculate the line integral e dxty2 dy in two ways: By parameterizing the pathC, and actually evaluating the integrals. . By finding a potential function for F(x,y)-+j and using the Fundamental theorem of Line integrals (which we'll discuss on Tuesday).
Suppose that is the path which starts at...
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)
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 point) Suppose F(x, y) = xyi + (x – y)j and C is the triangle from (4,0) to (-4,0) to (0,4) to (4,0). (a) Find the line integral of Ể along each segment of the triangle. Along C1, the line segment from (4,0) to (-4,0), the line integral is Along C2, the line segment from (-4,0) to (0,4), the line integral is Along C3, the line segment from (0,4) to (4,0), the line integral is (b) Find the circulation...
(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
5. Evaluate the integral c (2x -y)dx + (x + 3y)dy along the path C: line segment from (0,0) to (3,0) and (3,0) to (3,3)
5. Evaluate the integral c (2x -y)dx + (x + 3y)dy along the path C: line segment from (0,0) to (3,0) and (3,0) to (3,3)
Use Green's Theorem to calculate the line integral f. 2xy dx + 2(x+y) dy, where C is the unit circle centered at the origin and it is counter-clockwise oriented. $c 2xy dx + 2(x + y) dy =
(1) Evaluate the following line integrals in R3. r +yds for C the line segment from (0, 1,0) to (1, 0,0) for C the line segment from (0,1,1) to (1,0,1). for C the circle (0, a cos t, a sin t) for O (iv) 2π, with a a positive constant. t for C the curve (cost +tsint,sint tcost, 0) for Osts v3 (Hint for (i): use the parametrization (z, y, z) = (t, 1-t, 0) for 0 1) t
(1)...
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 +...
13. (6 pts) FTLIs, Green's, and Divergence Theorems (a) Complete the table below. Theorem Need to check: FTLIs The vector field Il curve Il surface IS: Green's Theorem | The vector field II curve ll surface is: and: Divergence Theorem The vector field |l curve l surface is: (b) For each of the following, choose all correct answers from the list below that can be used to evaluate the given integral. List items may be used more than once. i....