4. Find a potential for the conservative vector field: Fler,y) = (c + m2 te tua)
Decide whether or not the vector field is a gradient field (i.e. is conservative). If it is conservative, find a potential function. (ii) F(x,y)-6ญ่-12xVJ (iv) F(x, y, z)-< ye", e + z,y >
Decide whether or not the vector field is a gradient field (i.e. is conservative). If it is conservative, find a potential function. (ii) F(x,y)-6ญ่-12xVJ (iv) F(x, y, z)-
(1 point) Determine whether the vector field is conservative and, if so, find the general potential function. F = (cos z, 2y!}, -x sin z) Q= +c Note: if the vector field is not conservative, write "DNE". (1 point) Show F(x, y) = (8xy + 4)i + (12x+y2 + 2e2y)j is conservative by finding a potential function f for F, and use f to compute SF F. dr, where is the curve given by r(t) = (2 sinº 1)i +...
Determine whether or not the vector field is conservative. If it is conservative, find a function f such that F = Vf. (If the vector field is not conservative, enter DNE.) F(x, y, z) = 4xyi + (2x2 + 10yz)j + 5y2k f(x, y, z) =
6. (4) (a) Is F(x, y, z) = <e'siny, e cosx, esiny > a conservative vector field? Justify. (4) (b) Is F incompressible? Explain. Is it irrotational? Explain. (8) (c) The vector field F(x,y,z)= < 6xy+ e?, 6yx²+zcos(y), sin(y)+xe?> is conservative. Find the potential function f. That is, the function f such that Vf= F. Use a process. Don't guess and check.
2. a) Find a potential of the vector field f(x, y) = (a2 +2xy - y2, a2 - 2ry - y2) b) Show that the vector field (e" (sin ry + ycos xy) +2x - 2z, xe" cos ry2y, 1 - 2x) is conservative.
4. (16 points) Determine which of the following vector fields is conservative, and construct a potential function for that field. (iii) H(x, y, z) = (y2 +22,22 + 1, 2yz)
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...
is a conservative vector field (on its implied domain)
a. Find its potential function
b. Find
where C is the curve shown below and given by the
vector equation
Solve using concepts of Vector Fields, Line Integrals,
and/or The Fundamental Theorem for Line Integrals.
+sec2 F dr (sin-t-2, cos-t-2, cos(nt) 式t) 3 2 2 1 2 1 0 2
+sec2
F dr
(sin-t-2, cos-t-2, cos(nt) 式t)
3 2 2 1 2 1 0 2
I know Graph 1 is not conservative and Graph 2 is conservative
but how can we find vector function F for Graph 2? Because F is
deliberately not given.
Project 1. Fundamental theorem of line integrals amenta al theorem of line integrals: if F is a In our course we learned the conservative vector field with potential f and C is a curve connecting point A to b, then F dr f(B) f(A). Moreover it happens if and only if...
Consider the vector field F2(x, y)-(-y,z) and the closed curve C which is the square with corners (-1,-1), (1,-1), (1,1), and (-1,1) and is traversed counter-clockwise starting at (-1,-1) (a) Compute the outward flux across the curve C by calculating a line integral. (b) Use an appropriate version of Green's Theorem to compute the above flux as a (c) Compute the circulation of the vector field around the curve by computing a line (d) Use an appropriate version of Green's...