Question

Evaluate the surface integralF dS for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward) orientation. F(x, y, z) = zem-3zexyj + xyk, S is the parallelogram of this exercise with upward orientation. Evaluate the surface integral. (x + y + z)ds, S is the parallelogram with parametric equations

Answer 1

To evaluate the surface integral, we first need to parametrize the given parallelogram surface S. Then, we can calculate the surface normal vector and the magnitude of the cross product of the tangent vectors to set up the surface integral. Finally, by substituting the given vector field into the surface integral equation and plugging in the parametric equations, we can evaluate the surface integral and find the flux of the vector field across the parallelogram surface.

Step-by-step explanation:

To evaluate the surface integral, we first need to parametrize the given parallelogram surface S. Let's denote the parametric equations of S as:

x(s, t) = a + bs + ct

y(s, t) = d + es + ft

z(s, t) = g + hs + it

where a, b, c, d, e, f, g, h, i are constants that depend on the orientation and size of the parallelogram.

Once we have the parametric equations, we can calculate the surface normal vector N(s, t) and the magnitude of the cross product of the tangent vectors r_s and r_t. Then, we can set up the surface integral:

∫F · N dS = ∫(F · (r_s × r_t)) ds dt

By substituting the given F(x, y, z) = z e-3zexyj + xyk into the surface integral equation and plugging in the parametric equations, we can evaluate the surface integral and find the flux of F across the parallelogram surface S.