Indefinite Integrals of Scalar and Vector Fields: A Path Independence Dilemma?

[Jhenrique]

Is possible to compute indefinite integrals of functions wrt its variables, but is possible to compute indefinite integrals of scalar fields and vector fields wrt line, area, surface and volume?

 

[Matterwave]

Yes...I feel like you keep asking this same question, and I keep giving you the same answer. Look at Stoke's theorem:

$$\int_\Omega d\omega = \oint_{\partial \Omega} \omega$$

If we know that $\eta=d\omega$, then $\omega$ can be thought of as an "anti-derivative" of $\eta$, we can use the above theorem to generate a type of indefinite integral! Sadly, we get back an lower dimensional integral instead, but that is all that one can hope for as far as "indefinite integrals" work. Also this only works with exact forms, because non-exact forms are not necessarily derivatives of something. There's no way to define an "anti-derivative" if the base object is not the derivative of something.

The closest form of what you want is the Gradient theorem, which I showed you in a previous thread:

$$\int_a^b \vec{\nabla}\phi\cdot d\vec{r}=\phi(b)-\phi(a)$$

 

[Jhenrique]

Yeah, but I'm talking about a indefinite integral of a vector field! This can exist? This make sense?

So, I'm thinking in something like this:

$$\\ \int \vec{f} \cdot d\vec{r} = \int \vec{\nabla}\phi\cdot d\vec{r}=\phi$$

 

[Matterwave]

You tell me. When do you think a path integral makes any sense if you don't specify a path? When do you think a surface integral makes any sense if you don't specify a surface?

 

[Jhenrique]

You tell me. When do you think a path integral makes any sense if you don't specify a path? When do you think a surface integral makes any sense if you don't specify a surface?

No make sense think in a line integral without the line of integration like no make think in the area of integration of a function without the limits of integration. However, make sense think in the indefinite integral of a function, so why would make sense think in the indefinite integral of a field too?

 

[Matterwave]

What's the path of integration for a single variable function? Think this one through. You might realize, that a path was chosen for you already without you even thinking about it.

 

[Jhenrique]

But, independent of exist a geometric interpretation, this equation is true:

$$\\ \int \vec{f} \cdot d\vec{r} = \int \vec{\nabla}\phi\cdot d\vec{r}=\phi$$

?

 

[Matterwave]

No, you have to specify the endpoints or else that equation is meaningless.

What is $\phi$? It's a scalar function right? So it's really $\phi(x,y,z)$. Now what values of x, y and z are in the $\phi$ on the right? Does it make any sense?

If I change the path of integration, do you agree that the left hand side of that equation will change? Will the right hand side change?

Even in regular integrals. do we say:

$$\int \frac{d f}{dx} dx = f$$

? No we do not. We say:

$$\int \frac{d f}{dx} dx = f+C$$

Why is that C there? Is it necessary? Use your critical thinking. What is the "C" in the equation you posted?

 

[Jhenrique]

I can't use a critical thinking over a subject that is almost completely obscure for me...

C is a initial condition, or a guaranteed that integral is covering all the set of possibles antiderivatives.

 

[Matterwave]

If you are unable to even think critically about a question, it is best to learn a bit more about the subject first before making conjectures. To be honest Jhenrique, not to discourage you or your love of science/math, but a lot of your conjectures seem to make no sense. Perhaps they can lead somewhere if you can at least formulate the questions in a cogent manner.

There's a big book by Arfken and Weber called Mathematical Methods for Physicists (I'm a physicist, so I learn from physics textbooks) that covers basically everything you're worried about. You can use it as a study guide or a reference text. It's quite comprehensive. However, it looks at things from a physicist's perspective, so it's not always 100% rigorous like a math text would be. You might also want to pick up a pure math book on the subject and see where that takes you as well.

 

[micromass]

Yeah, but I'm talking about a indefinite integral of a vector field! This can exist? This make sense?

No, it doesn't. So if you want to discuss this further, you must provide a reference where this thing is defined.

 

[Jhenrique]

If you are unable to even think critically about a question, it is best to learn a bit more about the subject first before making conjectures. To be honest Jhenrique, not to discourage you or your love of science/math, but a lot of your conjectures seem to make no sense. Perhaps they can lead somewhere if you can at least formulate the questions in a cogent manner.

There's a big book by Arfken and Weber called Mathematical Methods for Physicists (I'm a physicist, so I learn from physics textbooks) that covers basically everything you're worried about. You can use it as a study guide or a reference text. It's quite comprehensive. However, it looks at things from a physicist's perspective, so it's not always 100% rigorous like a math text would be. You might also want to pick up a pure math book on the subject and see where that takes you as well.

When is about vector and tensor calculus my doubts are exponentially big... :(

More one thing... indefinite integrals requires (implicitly or not) path independence? In other words, only is possible to compute indefinite integrals of exact form?

 

[micromass]

When is about vector and tensor calculus my doubts are exponentially big... :(

More one thing... indefinite integrals requires (implicitly or not) path independence? In other words, only is possible to compute indefinite integrals of exact form?

Like I said, there is no such thing as indefinite path integrals.

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