Q.54 Let F1, F2, and F3 be functions of (x, y, z). Suppose that for every given pair of points A and B in space, the line integral...
Q.54 Let F1, F2, and F3 be functions of (x, y, z). Suppose that for every given pair of points A and B in space, the line integral
∫C (F1dx + F2dy + F3dz)
evaluates to the same value along any path C that starts at A and ends at B. Then which of the following is/are true?
F1 = ∂f/∂x, F2 = ∂f/∂y, F3 = ∂f/∂z.
Solution
The statement says that for every pair of points A and B, the line integral
∫C (F1dx + F2dy + F3dz)
has the same value for every path joining A and B. Therefore, the vector field
F = (F1, F2, F3)
is a conservative vector field (path-independent).
(A)
For any closed path Γ,
∮Γ (F1dx + F2dy + F3dz) = 0
A closed path starts and ends at the same point. Since the line integral is path-independent, the value of the integral around any closed curve must be zero.
True
(B)
A conservative vector field can be expressed as the gradient of a scalar potential function f(x, y, z).
F = ∇f
Therefore,
F1 = ∂f/∂x, F2 = ∂f/∂y, F3 = ∂f/∂z
True
(C)
This statement claims that
∂F1/∂x + ∂F2/∂y + ∂F3/∂z = 0
which means the divergence of F is zero:
∇·F = 0
However, a conservative field does not necessarily have zero divergence.
Consider:
f = x² + y² + z²
Then
F = (2x, 2y, 2z)
and
∇·F = 2 + 2 + 2 = 6 ≠ 0
False
(D)
For a conservative vector field,
∇ × F = 0
Therefore, the mixed partial derivatives satisfy:
∂F3/∂y = ∂F2/∂z
∂F1/∂z = ∂F3/∂x
∂F2/∂x = ∂F1/∂y
True
Final Answer
The correct options are:
(A), (B) and (D)
Answer: ABD