Question

Let C be the counter-clockwise planar circle with center at the origin and radius r₀. Without computing them, determine for the following vector fields whether the line integrals are positive, negative, or zero and type p, n, or z as appropriate.

a. Positive
b. Negative
c. Zero
d. Not mentioned

Answer 1

The sign of the line integral around a counter-clockwise path depends on the nature of the vector field; it is positive if the field is conservative or tangent in the counter-clockwise direction, negative if it opposes the motion, and zero if orthogonal or symmetric about the origin.

Step-by-step explanation:

The question involves determining the sign of the line integral of various vector fields around a closed path, specifically the planar circle C in the counter-clockwise direction. In general, if a vector field is conservative, or if the vector field is always tangent to the circle and follows the counter-clockwise direction, the line integral is positive. If the vector field opposes the motion (directed radially inward or tangent but in the clockwise direction), the integral would be negative.

If a vector field is orthogonal to every tangent vector along C, or if the field is symmetric with respect to the origin, then the line integral would be zero due to symmetry. It is also possible that a field is not explicitly mentioned, in which case we cannot determine the sign of the integral without additional information.