# 1-Forms .... Bachman, Exercise 3.2, Part 2 ....

• MHB
• Math Amateur
In summary: Plugging this back into the original differential form, we have:ω = df = (x^2y + h(y))dxBut since h(y) is a function of y only, we can ignore it when taking the derivative with respect to x. Therefore, we have:ω = 2xydx + h'(y)dx = (2xy + h'(y))dxNow, we can compare this to the given differential form ω = 2xydx + (x^2 - 2y)dy and equate the coefficients of dx. This gives us:2xy = 2xy + h'(y)h'(y) = 0
Math Amateur
Gold Member
MHB
I am reading David Bachman's book: "A Geometric Approach to Differential Forms" (Second Edition) ...

I need some help with Exercise 3.2, Part 2 ... Exercise 3.2, Part 2 ... reads as follows:View attachment 8607Bachman gives he answer to Exercise 3.2, Part 2 as $$\displaystyle dy = -4dx$$ ... ... but ... I cannot understand how he gets this answer ..Can someone please help ...

Peter=========================================================================================It may help MHB
readers of the above post to have access to Bachman's Section 3.2 ... ... so I am providing the same .. ... as follows:View attachment 8608
View attachment 8609
View attachment 8610It may also help MHB
readers of the above post to have access to Bachman's Section 3.1 ... ... so I am providing the same .. ... as follows:View attachment 8611
View attachment 8612
Hope that helps ...

Peter

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• Bachman ... Exercise 3.2 .png
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• Bachman - 1 - Section 3.1 - PART 1 ... .png
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• Bachman - 2 - Section 3.1 - PART 2 ... .png
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• Bachman - 3 - Section 3.1 - PART 3 ... .png
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• Bachman - 1 - Section 3.0 - PART 1 ... .png
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• Bachman - 2 - Section 3.0 - PART 2 ... .png
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Hello Peter,

Thank you for reaching out for help with Exercise 3.2, Part 2 from Bachman's book. I have taken a look at the section and I believe I can provide some assistance.

In this exercise, we are given the differential form ω = 2xydx + (x^2 - 2y)dy and we are asked to find the exact differential of ω. To do this, we need to use the definition of an exact differential, which states that a differential form ω is exact if and only if there exists a function f such that ω = df.

In order to find f, we need to integrate the given differential form ω. To do this, we need to use the properties of exact differentials, which state that the integral of an exact differential along a path from point a to point b is independent of the path chosen and depends only on the endpoints a and b.

Using this property, we can integrate ω along two different paths, one from (0,0) to (x,y) and another from (0,0) to (x,0). This gives us:

∫ω = ∫(2xydx + (x^2 - 2y)dy) = f(x,y) - f(x,0) = f(x,y) - f(0,0)

But since ω is an exact differential, this integral should be independent of the path chosen. Therefore, we can equate the two integrals and solve for f(x,y):

f(x,y) - f(0,0) = f(x,y) - f(x,0)

f(x,0) = f(0,0)

This means that f(x,y) is a constant with respect to y. Therefore, we can write f(x,y) = g(x) where g(x) is some function of x.

Now, we can use the definition of an exact differential again to find g(x). We have:

df = d(g(x)) = g'(x)dx

Comparing this with ω = 2xydx + (x^2 - 2y)dy, we can see that g'(x) = 2xy and g(x) = x^2y + h(y) where h(y) is some function of y.

Substituting this into f(x,y) = g(x), we have:

f(x,y)

## 1. What are 1-forms?

1-forms are mathematical objects used in multivariate calculus and differential geometry. They are a type of differential form, which is a generalization of the concept of a function. In simple terms, a 1-form is a linear function that takes in a vector as input and outputs a scalar value.

## 2. How are 1-forms used in science?

1-forms are used in various fields of science, such as physics, engineering, and computer science. They are particularly useful in the study of differential equations and vector fields. They are also used in the formulation of physical laws and equations, such as Maxwell's equations in electromagnetism.

## 3. What is the significance of Exercise 3.2, Part 2 in Bachman's work?

Exercise 3.2, Part 2 is a specific problem in Bachman's book that deals with the concept of 1-forms and their applications. It is an important exercise as it helps readers understand the properties and use of 1-forms in a practical setting.

## 4. Can you provide an example of a 1-form?

One example of a 1-form is the work done by a force on a moving object. This can be represented as a linear function that takes in the velocity of the object as input and outputs the amount of work done. In this case, the 1-form is a measure of energy.

## 5. Are 1-forms related to other mathematical concepts?

Yes, 1-forms are closely related to other mathematical concepts such as vector fields, differential equations, and tensors. They are also used in the study of differential geometry and topology. Understanding 1-forms can provide a deeper understanding of these other mathematical concepts as well.

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