State Fundamental theorem of Gradient, Divergence and Stokes.

The Fundamental Theorem of Gradient, Divergence, and Stokes are important concepts in vector calculus. These theorems relate the behavior of vector fields to the properties of their underlying functions.

1. The Fundamental Theorem of Gradient: This theorem states that if a function f(x, y, z) is continuously differentiable in a region of three-dimensional space, then the line integral of its gradient along a curve C is equal to the difference in the values of the function at the endpoints of the curve. Mathematically, it can be written as:

∫C ∇f · dr = f(P) - f(Q)

where ∇f is the gradient of f, C is the curve, P and Q are the endpoints of the curve, and dr is the differential displacement vector along the curve.

2. The Fundamental Theorem of Divergence: This theorem relates the flux of a vector field F through a closed surface S to the divergence of F within the region enclosed by S. Mathematically, it can be expressed as:

∫∫S F · dA = ∫∫∫V ∇ · F dV

where F is the vector field, S is the closed surface, dA is the differential area vector on S, ∇ · F is the divergence of F, V is the region enclosed by S, and dV is the differential volume element within V.

3. The Fundamental Theorem of Stokes: This theorem establishes a relationship between the circulation of a vector field F around a closed curve C and the curl of F within the region bounded by C. It can be stated as:

∫C F · dr = ∫∫S (∇ × F) · dA

where F is the vector field, C is the closed curve, dr is the differential displacement vector along C, (∇ × F) is the curl of F, S is any surface bounded by C, and dA is the differential area vector on S.

These fundamental theorems are widely used in various fields of physics and engineering to analyze and solve problems involving vector fields and their associated functions.