# Problem understanding the differential form of the circulation law

• havarija
In summary, the conversation discusses the definition of the curl of a vector field and its components. It is mentioned that if the dot product of the curl with unit vectors i, j, and k is equal to zero, then the curl of the field is also equal to zero. This leads to a discussion about the definition of the u-component of curl, where u is a unit vector.
havarija
I've encountered a problem in learning about the curl of a vector field.
(My learning material is the "Div, Grad, Curl and all that" from H.M. Shey.)

Introduction to problem:

The curl of a field F is defined as:
∇×F = i (∂Fz/∂y - ∂Fy/∂z) + j(∂Fx/∂z - ∂Fx/∂x) + k(∂Fy/∂x - ∂Fx/∂y)

He claims the following:
If we take:
n.(∇xF) for n = i, j, k
and they all equal 0 that we can conclude that ∇xF = 0 generally.

Is it not that we can only conclude that:
∂Fz/∂y = ∂Fy/∂z
∂Fx/∂z = ∂Fx/∂x and
∂Fy/∂x = ∂Fx/∂y

Or does this conclusion imply the following somehow?

Thanks .)

welcome to pf!

hi havarija! welcome to pf!

∂Fz/∂y = ∂Fy/∂z
∂Fx/∂z = ∂Fx/∂x and
∂Fy/∂x = ∂Fx/∂y

is the same as ∇xF = 0

(and, for any vector V, if n.V = 0 for n = i, j, k then V must be 0 !)

Now it seems like a ridiculous question :D

Silly me. But now at least I joined the forum :)

Welcome! I love ridiculous questions- I can actually answer some of them!

havarija said:
The curl of a field F is defined as:
∇×F = i (∂Fz/∂y - ∂Fy/∂z) + j(∂Fx/∂z - ∂Fx/∂x) + k(∂Fy/∂x - ∂Fx/∂y)

if youre reviewing it may be helpful to consider the following definition of curl component:

$$curlF\cdot u=\lim_{A(C)\to0}\frac{1}{A(C)} \oint_C F\cdot dr$$ where C is a closed loop, A(C) is the area of the loop, vector field F and unit vector u. this is the definition of the u-component of curl.

## 1. What is the differential form of the circulation law?

The differential form of the circulation law, also known as Stokes' theorem, states that the circulation of a vector field around a closed curve is equal to the surface integral of the curl of the vector field over the surface enclosed by the curve.

## 2. Why is the differential form of the circulation law important?

This law is important because it allows us to calculate the circulation of a vector field without having to explicitly calculate the line integral. It also provides a connection between line integrals and surface integrals, which can be useful in various applications.

## 3. How is the differential form of the circulation law derived?

The differential form of the circulation law is derived using vector calculus and the fundamental theorem of calculus. It can also be derived from the more general Gauss' theorem, which relates line integrals to volume integrals.

## 4. What are some real-world applications of the differential form of the circulation law?

The differential form of the circulation law has various applications in physics and engineering, such as in fluid dynamics, electromagnetism, and aerodynamics. It is also used in the study of weather patterns and ocean currents.

## 5. Are there any limitations to the differential form of the circulation law?

While the differential form of the circulation law is a powerful tool in vector calculus, it does have some limitations. It can only be applied to conservative vector fields, and it is not valid for non-continuous or non-differentiable vector fields.

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