What is the one-dimensional counterpart to the Green-Gauss theorem?

[Mankoo]

Homework Statement

a) In a three-dimensional situation, the spatial variation of a scalar field is given by the gradient. What is the one-dimensional counterpart?

b) In a three-dimensional situation, a volume integral of a divergence of a vector field can be transformed into a surface integral (Gauss’s theorem). What is the one-dimensional counterpart?

c) What is the one-dimensional counterpart to the Green-Gauss theorem?

Relevant Equations

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Are my answers to a and b correct?

a) In a three-dimensional situation, the spatial variation of a scalar field is given by the gradient. What is the one-dimensional counterpart? Answer:The derivative

b) In a three-dimensional situation, a volume integral of a divergence of a vector field can be transformed into a surface integral (Gauss’s theorem). What is the one-dimensional counterpart? Answer: Integration by parts

I have searched for this question but I can't find the right answer to question c. Can anyone help me?

c)What is the one-dimensional counterpart to the Green-Gauss theorem?

 

[Orodruin]

a) Yes.
b) No. I would say that integration by parts would correspond to Green’s identities (which can be derived as corrollaries from Gauss’ theorem).

 

[Mankoo]

So the answer for b is also the derivative?
what about c?

 

[FactChecker]

b) In a three-dimensional situation, a volume integral of a divergence of a vector field can be transformed into a surface integral (Gauss’s theorem). What is the one-dimensional counterpart? Answer: Integration by parts

What is the "surface" of an interval? What is the one-dimensional divergence? What theorem relates the values at the "surface" to the "divergence" in the interior of the interval?

 

[Mankoo]

I find this information on the website
The divergence theorem is an important result for the mathematics of physics and engineering, particularly in electrostatics and fluid dynamics. In these fields, it is usually applied in three dimensions. However, it generalizes to any number of dimensions. In one dimension, it is equivalent to integration by parts. In two dimensions, it is equivalent to Green's theorem.
Really don’t know the answer.

 

[Orodruin]

I find this information on the website
The divergence theorem is an important result for the mathematics of physics and engineering, particularly in electrostatics and fluid dynamics. In these fields, it is usually applied in three dimensions. However, it generalizes to any number of dimensions. In one dimension, it is equivalent to integration by parts. In two dimensions, it is equivalent to Green's theorem.
Really don’t know the answer.

What website? You have not given any reference.

Also, you need to provide your reasoning. Why do you think partial integration would correspond to Gauss’ theorem? You cannot just guess wildly or search for the answer. You need to reason.

 

[FactChecker]

In one dimension, it is equivalent to integration by parts.

It loses me there. I don't see it. I see it as related to a much more fundamental fact.

 

[Orodruin]

It loses me there. I don't see it. I see it as related to a much more fundamental fact.

It loses you because it is simply not correct. As I mentioned in #2:

integration by parts would correspond to Green’s identities (which can be derived as corrollaries from Gauss’ theorem).

Edit: It should also be mentioned that integration by parts may be derived from the answer to b and the Leibniz rule for the first derivative.

 

[Orodruin]

So the answer for b is also the derivative?
what about c?

And no, it is not ”derivative” either. It is a very particular theorem relating the integral of some sort of derivative of something over an interval to the value of that something on the boundary of the interval.

 

[FactChecker]

We have given some very strong hints. I suggest that you give more thought to those hints.

 

[Mankoo]

So is my answer correct now?
a) In a three-dimensional situation, the spatial variation of a scalar field is given by the gradient. What is the one-dimensional counterpart? Answer:The flux integral of v over a bounding surface is the integral of its divergence over the interior.

b) In a three-dimensional situation, a volume integral of a divergence of a vector field can be transformed into a surface integral (Gauss’s theorem). What is the one-dimensional counterpart? answer: differential equation

c) What is the one-dimensional counterpart to the Green-Gauss theorem? Integration by parts

 

[pasmith]

What is the 1D equivalent of a volume? An interval.
What is the 1D equivalent of the surface of a volume? The end points of the interval.
What is the 1D equivalent of the divergence? The derivative.

What theorems from calculus in a single real variable do we have relating the values of a function at the end points of an interval to the integral of its derivative?

 

[Mankoo]

Differential equations?
Is a and c correct answer?

 

[FactChecker]

Hint: It's a very fundamental theorem in the field of calculus. :-)

 

[Mankoo]

a) In a three-dimensional situation, the spatial variation of a scalar field is given by the gradient. What is the one-dimensional counterpart? Answer: The derivative

b) In a three-dimensional situation, a volume integral of a divergence of a vector field can be transformed into a surface integral (Gauss’s theorem). What is the one-dimensional counterpart? answer: The flux integral of v over a bounding surface is the integral of its divergence over the interior.

c) What is the one-dimensional counterpart to the Green-Gauss theorem? Integration by parts

Which question is wrong?

 

[Mankoo]

Hint: It's a very fundamental theorem in the field of calculus. :-)

so the answer for b is: Integrals and antiderivatives ?

 

[Orodruin]

I am sorry, but it seems to me that you are guessing wildly rather than actually pondering the hints given to you. I suggest you take a deep breath, take a step back, reread the thread, and consider the hints that have been dropped.

Particularly, consider this:

What is the 1D equivalent of a volume? An interval.
What is the 1D equivalent of the surface of a volume? The end points of the interval.
What is the 1D equivalent of the divergence? The derivative.

What theorems from calculus in a single real variable do we have relating the values of a function at the end points of an interval to the integral of its derivative?

and what the divergence theorem actually says.

 

[Orodruin]

What is the one-dimensional counterpart? answer: The flux integral of v over a bounding surface is the integral of its divergence over the interior.

For example, what you have stated here is just the divergence theorem itself … not its 1D counterpart.

 

[FactChecker]

The flux integral of v over a bounding surface is the integral of its divergence over the interior.

Go way back to your first calculus course. What theorem relates the integral of one function over an interval to the values of another function at the endpoints? Do you see the similarity of that to these theorems in higher dimensions?

 

[Mankoo]

I’m trying and the calculus book is not really helping.

Which question is wrong?

a)In a three-dimensional situation, the spatial variation of a scalar field is given by the gradient. What is the one-dimensional counterpart? Answer: The derivative

b) In a three-dimensional situation, a volume integral of a divergence of a vector field can be transformed into a surface integral (Gauss’s theorem). What is the one-dimensional counterpart? answer: The gradient theorem

c) What is the one-dimensional counterpart to the Green-Gauss theorem? Integration by parts

 

[Orodruin]

I’m trying and the calculus book is not really helping.

Which question is wrong?

a)In a three-dimensional situation, the spatial variation of a scalar field is given by the gradient. What is the one-dimensional counterpart? Answer: The derivative

b) In a three-dimensional situation, a volume integral of a divergence of a vector field can be transformed into a surface integral (Gauss’s theorem). What is the one-dimensional counterpart? answer: The gradient theorem

c) What is the one-dimensional counterpart to the Green-Gauss theorem? Integration by parts

Again, you are just guessing. That is not constructive and significantly decreases the possibility for us to actually help you. You need to provide an argument, not just guess wildly.

 

[Mankoo]

Again, you are just guessing. That is not constructive and significantly decreases the possibility for us to actually help you. You need to provide an argument, not just guess wildly.

I really don’t understand and I have been reading the calculus book, but can’t find the answer. I have been searching for all the hint I get from you but I still can’t find the answer.

And which question is wrong?

 

[Orodruin]

I really don’t understand and I have been read the calculus book, but can’t find the answer. I have been searching for all the hint I get from you but I still can’t find the answer.

And which question is wrong?

You gave an attempted answer. The question we are asking you is simple: Why did you give those answers? By answering that question you will both help yourself by having to ponder and express your reasoning and you will help us understand your thought process and correct it if wrong. It is much more educational than us simply blurting out the answers but it requires you to actually engage with us rather than continuously just providing an answer without any reasoning and asking if it is correct.

 

[FactChecker]

Ok, although we are only supposed to give hints and guidance for homework-type problems, I can't resist any longer. Look up the Fundamental Theorem of Calculus.
If the derivative of $F(x)$ is $f(x)$, then $\int_a^b f(x) dx = F(b) - F(a)$.
Look at @pasmith 's post #12 and you should see a striking similarity between this and the higher-dimensional theorems.

 

[WWGD]

I really don’t understand and I have been reading the calculus book, but can’t find the answer. I have been searching for all the hint I get from you but I still can’t find the answer.

And which question is wrong?

You can look up Divergence Theorem in Wikipedia:

 

[Mankoo]

You can look up Divergence Theorem in Wikipedia:

I did look up that on wikipedia, that why I wrote integration by parts. But you are saying it’s wrong.
So I really don’t know.

 

[Orodruin]

You can look up Divergence Theorem in Wikipedia:

He did and it is wrong (as already pointed out in this thread). Wikipedia is not always to be trusted.

 

[WWGD]

You can look up Divergence Theorem in Wikipedia:

He did and it is wrong (as already pointed out in this thread). Wikipedia is not always to be trusted.

Ok, didn't read it too carefully. I'm accessing from my phone, which makes it harder.

 

[Mankoo]

This should be the answer for b? or I still wrong?