Question

If curl F = (x² + z²)j; + 10k, find ∫F. dr, where C is a circle of radius 7, centered at the origin, with the following conditions: NOTE: Enter exact answers, or round to three decimal places. (a) C in the xy-plane, oriented counterclockwise when viewed from above. (b) C in the xz-plane, oriented counterclockwise when viewed from the positive y-axis.

Answer 1

1. For C in the xy-plane: ∫F ⋅ dr = 0.

2. For C in the xz-plane: ∫F ⋅ dr = 0.

Explanation:

For the circle C in the xy-plane, the vector field F is given by (x² + z²)j + 10k. As the circle is oriented counterclockwise in the xy-plane, the radial vector dr is parallel to the xy-plane. The dot product between F and dr involves only the k-component of F, which is 10k. Since dr is perpendicular to this component, the dot product becomes zero. Consequently, the line integral ∫F ⋅ dr over C in the xy-plane evaluates to zero.

Similarly, when the circle C is in the xz-plane, F has components along the j and k directions. However, the radial vector dr is parallel to the xz-plane. Again, the dot product between F and dr is zero as the j-component of F does not contribute to the line integral. Therefore, ∫F ⋅ dr for C in the xz-plane also equals zero.

These results signify that the vector field F does not induce any circulation or flow along the specified circles in the given planes. The perpendicular orientation of the radial vector dr with respect to the relevant components of F in each case ensures that the dot product is null, leading to a line integral of zero.