Physical problem of stoke’s theorem

1.     Purpose: Stokes’ Theorem generalizes Green’s Theorem to three dimensions.

2.     Concept: It relates a surface integral over a smooth surface (S) to a line integral around the boundary of that surface.

3.     Curl and Circulation: The theorem connects the concept of curl (rotational behavior) with circulation (line integral).

4.     Flux and Boundary: We can calculate the flux of the curl of a vector field across (S) using information only along the boundary of (S).

5.     Mathematical Form: For a vector field (\mathbf{F}), the theorem states:
(\oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_S \text{curl}(\mathbf{F}) \cdot d\mathbf{S})

6.     Orientation: The orientation of (S) induces the positive orientation of the boundary curve (C).

7.     Applications: Stokes’ Theorem has applications in fluid mechanics, electromagnetism, and more.

Maxwell-Faraday Equation: It helps derive important physics equations, like Faraday’s law

Question:1

Let's consider a physical example of Stokes' theorem in the context of fluid dynamics and fluid circulation.

Imagine you have a flowing river, and you want to calculate the circulation of the fluid around a closed loop in the river. The circulation represents the flow of the fluid around the loop, which can be thought of as the net "rotational" effect of the fluid motion.

Now, let's define the physical quantities involved:

1. The vector field: In this case, the vector field represents the velocity of the fluid at each point in space. Let's call it V(r), where r represents the position vector in the river.

2. The closed loop: We'll consider a closed curve C that lies on a plane within the river.

Stokes' theorem relates the circulation of the fluid around the closed curve C to the curl of the vector field V(r) integrated over a surface S that the loop C bounds. The theorem can be stated as follows:

∮(C) V(r) · dr = ∬(S) (curl V) · dA

Here's a step-by-step explanation using the physical example:

1. Choose a closed loop: Consider a circular loop C in the river, which lies on a flat surface of the water.

2. Calculate the circulation: Integrate the velocity vector field V(r) along the closed loop C. This means summing up the component of velocity tangential to the curve at each point on the loop as you move around the curve.

3. Calculate the curl of V: Calculate the curl of the velocity vector field V(r) at each point in space. The curl represents the local rotation or vorticity of the fluid at each point.

4. Choose a surface bounded by the loop: Select a surface S that lies in the plane of the loop C and is bounded by the loop C.

5. Calculate the surface integral: Integrate the curl of the vector field V(r) over the surface S. This involves taking the dot product of the curl of V with the area vector dA at each point on the surface and summing them up.

According to Stokes' theorem, the circulation of the fluid around the closed loop C (left-hand side of the equation) is equal to the surface integral of the curl of the velocity vector field V(r) over the surface S (right-hand side of the equation). This theorem shows the deep connection between fluid circulation and the local rotation of the fluid in the surrounding region.

Example:

Use Stokes’ Theorem to evaluate 1C F # dr, if F = xzi + xyj + 3xzk and C is the boundary of the portion of the plane 2x + y + z = 2 in the first octant, traversed counterclockwise as viewed from above (Figure 16.67)

Solution The plane is the level surface ƒ(x, y, z) = 2 of the function ƒ(x, y, z) = 2x + y + z. The unit normal vector

n = ∇ƒ / |∇ƒ|  = (2i + j + k) / |2i + j + k| = 1 / (2i + j + kb)

is consistent with the counterclockwise motion around C. To apply Stokes’ Theorem, we find

curl F = ∇ * F =

 = (x - 3z)j + yk

On the plane, z equals 2 - 2x - y, so ∇ * F = (x - 3(2 - 2x - y))j + yk = (7x + 3y - 6)j + yk

And 

 (∇ * F) # n = 1/  (7x + 3y - 6 + y) = 1 / (7x + 4y – 6)

The surface area differential is

 ds = | ∇ƒ|  /| ∇ƒ . k|dA = dx dy

The circulation is

  F . dr = ∫∫S(∇ * F) . n da

=   (7x + 4y – 6)  dy dx

=  (7x + 4y - 6) dydx = -1

Question:2

In fluid dynamics, a circular whirlpool is formed by a rotating body of water. Consider a scenario where the velocity field of the whirlpool is given by V = (2xy)i + (x² - z)j + (y² + z)k in three-dimensional space. The whirlpool is bounded by a circular path in the xy-plane. How can we use Stokes' theorem to determine the circulation of the fluid flow around the circular path?

 

Answer:

To determine the circulation of the fluid flow around the circular path using Stokes' theorem, we can follow these steps:

 

1. Calculate the curl of the velocity field V, which is given by curl V = (∂Vz/∂y - ∂Vy/∂z)i + (∂Vx/∂z - ∂Vz/∂x)j + (∂Vy/∂x - ∂Vx/∂y)k.

 

2. Evaluate the line integral of the velocity field V along the circular path. This involves integrating V · dr, where dr represents the differential displacement along the circular path.

 

3. Alternatively, we can calculate the surface integral of the curl of V over the surface bounded by the circular path in the xy-plane. The surface integral involves integrating the dot product of the curl of V and the outward-pointing normal vector dS to the surface.

 

By applying Stokes' theorem, we can equate the line integral of V along the circular path to the surface integral of the curl of V over the surface bounded by the path.

 

Therefore, by utilizing Stokes' theorem, we can determine the circulation of the fluid flow around the circular path by either directly calculating the line integral or evaluating the surface integral using the curl of the velocity field.