Question

The gradient of a scalar field \( S(x, y, z) \) has the following characteristic(s).

• A. Line integral of a gradient is path-independent
• B. Closed line integral of a gradient is zero
• C. Gradient of \( S \) is a measure of the maximum rate of change in the field \( S \)
• D. Gradient of \( S \) is a scalar quantity

Hint:

The gradient of a scalar field is a vector that points in the direction of the maximum rate of change of the field.

Answer 1

The Correct Option is A, B, C

Step 1: Understanding the properties of a gradient.
The gradient of a scalar field \( S(x, y, z) \) gives the direction of the maximum rate of increase of the field and its magnitude represents the rate of change in that direction. The gradient is a vector quantity and points in the direction of the steepest ascent of the scalar field.

Step 2: Analyzing the options.
(A) Line integral of a gradient is path-independent: Incorrect. This is true for conservative vector fields, but not a unique characteristic of a gradient itself.
(B) Closed line integral of a gradient is zero: Incorrect. This is true for a conservative field, but not a general property of gradients.
(C) Gradient of \( S \) is a measure of the maximum rate of change in the field \( S \): Correct. The gradient of a scalar field gives the direction of maximum change.
(D) Gradient of \( S \) is a scalar quantity: Incorrect. The gradient of a scalar field is a vector quantity, not a scalar.

Step 3: Conclusion.
The correct answer is (C) because the gradient of a scalar field represents the direction and magnitude of the maximum rate of change.