Diferentiation and differential operators

• Jhenrique
In summary, the gradient of f can be expressed as \nabla f = (\bf{d} f)^\sharp, the curl of f can be written as \nabla\times \bf{f}=[\star(\bf{d}\bf{f}^\flat)]^\sharp, and the divergence of f is given by \nabla\cdot \bf{f}=\star\bf{d}(\star \bf{f}^\flat). These operations are all related to the exterior derivative and can be explained using the musical isomorphisms and Hodge dual operator. The term "fractional differentiation" has a different meaning and definition in mathematics.
Jhenrique
If the gradient of f is equal to differential of f wrt s: $$\vec{\nabla}f=\frac{df}{d\vec{s}}$$ so, what is the curl of f and the gradient of f in terms of fractional differentiation?

I'm not sure what you mean by "fractional differentiation" (wikipedia has a definition that seems very different than what you're talking about here), but these three operations are all related to the exterior derivative (if that's what you were talking about) $\bf{d}$.

$$\nabla f = (\bf{d} f)^\sharp$$
The Curl
$$\nabla\times \bf{f}=[\star(\bf{d}\bf{f}^\flat)]^\sharp$$
The Divergence
$$\nabla\cdot \bf{f}=\star\bf{d}(\star \bf{f}^\flat)$$

Edit: I have no idea why the f is boldfaced in the gradient formula...and I don't know how to fix it. It should be non-boldfaced as it's a functinon.

Last edited:
Matterwave said:
Edit: I have no idea why the f is boldfaced in the gradient formula...and I don't know how to fix it. It should be non-boldfaced as it's a functinon.

Hmm, this is weird. It looks like a bug, thanks for bringing this to our attention.

Oh I should have explained my notation. The sharp and flat signs mean the musical isomorphisms which, given a metric, maps a one form to a vector and vice versa. The star is the Hodge dual operator which, again being metric dependent, maps a (k)-form to a (n-k) form where n is the dimension of the manifold.

Matterwave said:
I'm not sure what you mean by "fractional differentiation" (wikipedia has a definition that seems very different than what you're talking about here), but these three operations are all related to the exterior derivative (if that's what you were talking about) $\bf{d}$.

Fractional differentiation means, for me, express a derivative in the form of a fraction.

Oh, so you mean like...

$$\nabla\cdot\vec{f}=\frac{\partial f_x}{\partial x}+\frac{\partial f_y}{\partial y}+\frac{\partial f_z}{\partial z}$$

?

Matterwave said:
Oh, so you mean like...

$$\nabla\cdot\vec{f}=\frac{\partial f_x}{\partial x}+\frac{\partial f_y}{\partial y}+\frac{\partial f_z}{\partial z}$$

?

Yes and not. Yes because you used fraction, and not because your fraction is a scalar.

A Hessian of f, for example, can be write like:

$$Hf = \frac{d^2f}{d\vec{r}^2}$$

Jhenrique said:
Fractional differentiation means, for me, express a derivative in the form of a fraction.
Then your definition is at odds with how the term is already used, which has nothing to do with the derivative appearing in the form of a fraction. See this article on fractional derivatives.

What is differentiation?

Differentiation is a mathematical process used to calculate the rate of change of a function at a specific point. It involves finding the derivative, or slope, of a function at a given point.

What are differential operators?

Differential operators are mathematical symbols or operations that are used to calculate derivatives. They are used to manipulate functions to find their derivatives, and they include operations such as differentiation, integration, and partial differentiation.

What is the difference between differentiation and integration?

Differentiation is the process of finding the derivative of a function, while integration is the process of finding the area under a curve. In other words, differentiation deals with rates of change, while integration deals with accumulation and finding the total value of a function.

How is differentiation used in real life?

Differentiation is used in various fields such as physics, economics, and engineering to analyze and model real-life situations. For example, it can be used to calculate the velocity of an object, the rate of change of a stock price, or the slope of a curve in a road design.

What are some common applications of differential operators?

Differential operators have many applications in mathematics, physics, and engineering. Some common applications include solving differential equations, finding extrema of functions, and modeling physical systems such as heat flow and fluid mechanics.

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