# Some questions about double integral and vector calculus

• abcdefg10645
In summary: For Q1:u=3x+y, v=x-2yFor Q2:u=3x+y, v=x-2y, dxdy= … ?For Q3:u=3x+y, v=x-2y, dxdy= … ?For Q4:u=3x+y, v=x-2yFor Q5:u=3x+y, v=x-2yFor Q6:u=3x+y, v=x-2yFor Q7:
abcdefg10645
I've written all the questions in the PDF file ...

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If you can't be bothered to
1) type the problem itself here
2) Do anything at all at all on the problem

I don't see why I should bother.

HallsofIvy said:
If you can't be bothered to
1) type the problem itself here
2) Do anything at all at all on the problem

I don't see why I should bother.

I'm sorry for that.

I scanned the problem I had just because they are "hard" to type ...I concede that my computer ability is not good... So I made manuscript instead.

And I did not discribed the problem I have ...is my false .

I'm now typing the problem I have...

For Q1:

I tried change of variable and fail...

I could not figure out the craftsmanship solving the problem...

For Q2:

I was considering expand the formula by assuming

A=(A1)i+(A2)j+(A3)k {also B}

substitude the equations...

No doubt ...It is quite terrible !

Sorry again for ignoring the "detail problem" :shy:

Hi abcdefg10645! That's better!

(have a del: ∇ and a dot: · and an integral: ∫ )

For 1, make the obvious substitution …

u = 3x + y, v = x - 2y, dxdy = … ?

For 2, just write each of the coordinates out in full …

try a. first …

what do you get?

tiny-tim said:
Hi abcdefg10645! That's better!

(have a del: ∇ and a dot: · and an integral: ∫ )

For 1, make the obvious substitution …

u = 3x + y, v = x - 2y, dxdy = … ?

For 2, just write each of the coordinates out in full …

try a. first …

what do you get?

But for question 1...I still have a little problem...

Could you give me a hint for changing the upper and lower limit?

And even tell me the "principle" resoving the limits ...

Thanks!

abcdefg10645 said:
Could you give me a hint for changing the upper and lower limit?

I don't understand

just convert x and y to u and v …

what do you get?

The region in problem 1 has boundaries 2x+ y= 0, 3x+ y= 0, x- 2y= 1 and x- 2y= 2.

tiny-tim suggested the substitution u = 3x + y, v = x - 2y.

Solve for x and y in terms of u and v: Multiplying the first equation by 2 and adding to the second equation gives 2u+ v= 6x+ 2y+ x- 2y= 7x so x= (2u+ v)/7. Multiplying the second equation by 3 and subtracting the first equation gives 3v- u= 3x-6y- 3x- y= -7y so y= (u- 3v)/7.

Convert the boundary to uv-coordinates by substituting those for x and y:
2x+ y= 2(2u+v)/7+ (u-3v)/7= (4u+ 2v+ u- 3v)/7= (5u- v)/7= 0 so 5u- v= 0

It should be obvious that the last three equations become u= 0, v= 1, and v= 2, but as a demonstration, x- 2y= (2u+ v)/7- 2(u- 3v)/7= (2u+ v- 2u+ 6v)/7= 7v/7= v= 1.

So you want to integrate on the region bounded by the lines u= 0, u= v/5, v= 1, and v= 2. I recommend taking the "outer integral" to be with respect to v, from 1 to 2, and the "inner integral" with respect to u, from 0 to v/5. Be careful about dudv. It is, of course, the Jacobian.

## 1. What is the definition of a double integral?

A double integral can be thought of as a mathematical tool used to calculate the volume under a surface in two-dimensional space. It involves evaluating the function at different points within a given region and summing up these values.

## 2. How is a double integral related to vector calculus?

Vector calculus involves the study of vector fields and their properties. Double integrals are used in vector calculus to calculate flux, which is the amount of fluid, energy, or particles that pass through a given surface. This is an important concept in understanding how vector fields behave.

## 3. What are the different types of double integrals?

There are two types of double integrals: iterated integrals and double integrals over a region. Iterated integrals involve integrating a function with respect to one variable at a time, while double integrals over a region involve integrating a function over a two-dimensional region in one step.

## 4. How do you set up a double integral?

To set up a double integral, you first need to determine the limits of integration, which are the boundaries of the region over which the integral will be evaluated. Then, you need to choose the order of integration, which is the order in which the variables will be integrated. Finally, you need to write the function to be integrated in terms of the chosen variables and set up the integral using the appropriate notation.

## 5. What are some applications of double integrals and vector calculus?

Double integrals and vector calculus have various applications in physics, engineering, and economics. They can be used to calculate the work done by a force, the flow of a fluid, the area of a surface, and the volume of a solid. They are also essential in understanding and solving problems related to electric and magnetic fields, fluid dynamics, and optimization.

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