Line integration - Path independence theorem

Q1. For a vector field V^→ = (2xy - z², x² + 4yz, 2y² - 2xz) calculate its line integral along each of the following curves:

a. Straight line segment from A(1, 2, -1) to B(2, 3, 1)

b. From A to B (the same points as above) but following a sequence of segments, each parallel to some coordinate axis. Like first along x, then along y, and then along z (or in any other order)

c. Circle or radius 2, center C(1, 2, 2), parallel to the yz-plane

d. Square with one corner at D(2, 2, 2), and opposite corner at E(5, 5, 2)

e. Main diagonal of a cube of edge 10 with one corner at origin and with edges along the axes, in the first octant.

f. From the origin, to the opposite corner of the same cube above, but following the diagonal of one cube and then one more edge.

g. Compare the results of the questions (a)...(f) above, make comments, and explain results.

2. Calculate curl V^→. Comment the result.

3. Show that if we choose a scalar field φ = x²y - z²x + 2y²z this is a potential for V^→.

4. Recalculate questions 1.a, 1.d, and 1.e using the potential and the fundamental theorem of calculus applied to path independence theorem.

5. For a vector field W^→→→ = (y - z, x + z, y - x) calculate its line integral along each of the following curves:

a. Straight line segment from D(3, -2, 4) to B(0, 0, 4)

b. From A to B (the same points as above) but following a sequence of segments, each parallel to some coordinate axis. Like first along x, then along y, and then along z (or in any other order)

c. Circle or radius 3, center C(-1, -2, -2), parallel to the xz-plane

d. Helix of radius 2, pitch 3, with 7 full turns, beginning at point G(2, 0, 0), directed vertical with axis along z-axis, and basis circle in xy-plane centered at origin.

e. Straight segment between the beginning point and end point of the helix above.

f. Straight line segment from D(12, 13, 14) to B(15, 16, 17)

g. Compare the results of the questions (a)...(f) above, make comments, and explain results.

6. Calculate curl W^→→→. Comment the result.

7. Show that if we choose a scalar field φ = xy + yz - xz this is a potential for W^→→→.

8. Recalculate questions 1.a, 1.d, and 1.f using the potential and the fundamental theorem of calculus applied to path independence theorem.

9. Show that the vector field r^r4 is potential. On what region of the physical space is it conservative?

Find constants a and b such that r_b is its potential.

10. Calculate line integral for r-^r 3 along a meridian of a origin-centered-sphere of radius 5, from north pole to south pole.

11. Prove that r·dr^→ = r dr

12. Calculate line integral for r-^r 3 from H(0, 0, 5) to J(0, 0, 6) along the z-axis. Use two methods: directly as a line integral, and by using the potential for this field (you need to find it). Hint: use problem 11 above.