Orthogonal Curvilinear Coordinates:

Orthogonal curvilinear coordinates are a set of coordinates that are defined in terms of curved surfaces and are mutually perpendicular. These coordinates are useful in describing physical phenomena occurring in systems with curved boundaries or in problems involving spherical or cylindrical symmetry.

In orthogonal curvilinear coordinates, the position vector can be written as:

r = f1(u1, u2, u3) * e1 + f2(u1, u2, u3) * e2 + f3(u1, u2, u3) * e3

where u1, u2, u3 are the curvilinear coordinates, and e1, e2, e3 are the unit vectors in the respective coordinate directions.

Gradient of a Scalar in Orthogonal Curvilinear Coordinates:

The gradient of a scalar function in orthogonal curvilinear coordinates can be obtained by taking the partial derivatives of the function with respect to each coordinate and multiplying them by the corresponding unit vectors.

Expression for Gradient:

Let's consider a scalar function φ(u1, u2, u3) in orthogonal curvilinear coordinates. The gradient of φ can be expressed as:

∇φ = ∂φ/∂u1 * e1 + ∂φ/∂u2 * e2 + ∂φ/∂u3 * e3

where ∂φ/∂u1, ∂φ/∂u2, and ∂φ/∂u3 are the partial derivatives of φ with respect to u1, u2, and u3, respectively.

Explanation:

To calculate the gradient, we need to take the partial derivatives of φ with respect to each coordinate. These derivatives represent the rate of change of the function in each direction.

The unit vectors e1, e2, and e3 are specific to the chosen coordinate system and are mutually perpendicular. They indicate the direction of each coordinate axis.

Multiplying the partial derivatives by the corresponding unit vectors gives us the components of the gradient vector. The gradient vector points in the direction of maximum increase of the scalar function.

Conclusion:

In summary, orthogonal curvilinear coordinates provide a useful framework for describing systems with curved boundaries or spherical/cylindrical symmetry. The gradient of a scalar function in this coordinate system can be obtained by taking partial derivatives with respect to each coordinate and multiplying them by the respective unit vectors. The resulting vector represents the direction and magnitude of maximum increase of the scalar function.