Question

Let C be the counter-clockwise planar circle with center at the origin and radius r > 0. Without computing them, determine for the following vector fields F whether the line integrals ∫_c F. dX are positive, negative, or zero and type P, N, or Z as appropriate. A. F = the radial vector field = x1 + yJ: = B. F = the circulating vector field = -y[ + xJ: C. F = the circulating vector field = yI – XJ: D. F = the constant vector field = I+J:

Answer 1

The line integrals will be positive for the radial vector field, zero for the circulating vector fields, and dependent on the path orientation for the constant vector field.

Step-by-step explanation:

To determine whether the line integrals ∫c F ⋅ dX are positive, negative, or zero for the given vector fields, we need to consider the direction of the field and the direction of the path along the circle.

For vector field A, which is the radial vector field, the line integral will be positive because the field is pointing outward from the origin and the path is moving counter-clockwise.

For vector fields B and C, which are circulating vector fields, the line integrals will be zero because the fields are tangent to the curve of the circle and there is no component of the field along the path.

For vector field D, which is the constant vector field, the line integral will depend on the orientation of the path along the circle. If the path is clockwise, the line integral will be negative, and if the path is counter-clockwise, the line integral will be positive.