Use stokes' theorem to evaluate c f · dr where c is oriented counterclockwise as viewed from above. f(x, y, z) = xyi + 5zj + 7yk, c is the curve of intersection of the plane x + z = 1 and the cylinder x2 + y2 = 81.

Step-by-step explanation:

The given curve is .

Let us parameterize C to obtain a vector equation for C.

We have ,   .

This is the equation of a circle( a cylinder in space) with radius, r=9 units.

The parametric equation of a circle is given by:

and

This implies that:

and

From, , we obtain, ,

A vector equation for C is;

for

Also, we have;

Since C is oriented counterclockwise as viewed from above, we can apply the Stokes' Theorem.

Thus, by Stokes' Theorem;

guess at question:

a vertical line intersects the graph of g(x,y) = 0 at two points. what conclusion follows about the function f(x) implicitly defined by g(x,f(x)) = 0?

the domain of f(x) cannot include the x coordinate of any such vertical line.

informally, f(x) is not a function, and

g(x,y)=0 cannot be solved for y uniquely.

the relation r = {(x,y) such that g(x,y) = 0} is a function if

for all (a,b) ∊ r and (c,d) ∊ r, a=c implies b=d.

step-by-step explanation: