Differential Form Homework on Unit Sphere in R3: Exactness?

In summary, the conversation discusses the form w=xdydz in R^3 and the unit sphere S^2. The question is whether w is exact when restricted on S^2. Two different approaches are suggested, one using stereographic projection and the other using the theorem about closed and exact forms. The conclusion is that w is not exact on S^2.
  • #1
daishin
27
0

Homework Statement


Let w be the form w= xdydz in R^3. Let S^2 be the unit sphere in R^3.
If we restrict w on S^2, is w exact?


Homework Equations





The Attempt at a Solution


My guess is w is not exact on S^2.
Suppose w is exact on S^2. Then w=da for some 1-form a=fdx+gdy+hdz.
Then by definition of exterior derivative, we get
w=(-df/dy+dg/dx)(dx^dy)+(-df/dz+dh/dx)(dx^dz)+(-dg/dz+dh/dy)(dy^dz)
So we get the conditions:
df/dy=dg/dx, df/dz=dh/dx, x=-dg/dz+dh/dy.
I think I should use a fact that I am working on a unit sphere. Could anybody help me?
 
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  • #2
I did not tried but it would be useful to write everything in spherical coordinates and to take radius as 1.
 
  • #3
streographic projection

Using stereographic projection,(say (s,t)) I attained follwoing condition if I assume xdydz is exact on S^2,

for some smooth function g and f,
-df/dt+dg/ds = (-24(s^2)(t^2)-8(s^4)-8s)/(1+s^2+t^2)^4.
Now in order to show xdydz is not exact, it suffices to show such f and g does not exist. How can I show it?
 
  • #4
Different way: I tried to use the thm saying if a differential form is exact then
its closed.
i.e.if you can show dw is not equal to zero(meaning not closed)
you can conclude w is not exact.If i didn't make a mistake dw=dx dy dz
But still don't now how it is related to S^2
 
  • #5
matness said:
Different way: I tried to use the thm saying if a differential form is exact then
its closed.
i.e.if you can show dw is not equal to zero(meaning not closed)
you can conclude w is not exact.If i didn't make a mistake dw=dx dy dz
But still don't now how it is related to S^2

dw IS zero when restricted to S^2, it's a three form on a two manifold. But you can't apply the Poincare lemma to show it's exact since S^2 isn't contractible.
 
  • #6
Try it in spherical coordinates, as matness suggested. It's pretty straight forward to find a solution to w=da, I think.
 

1. What is a differential form?

A differential form is a mathematical object used in multivariable calculus and differential geometry to represent the concept of integration over a manifold. It is a combination of functions and their derivatives, which allows for the calculation of integrals over curved surfaces or higher-dimensional spaces.

2. What is a unit sphere in R3?

A unit sphere in R3, also known as a 3-sphere, is a three-dimensional geometric shape that is the set of all points in three-dimensional space equidistant from a central point. It can be thought of as a three-dimensional version of a circle, with points on the surface of the sphere having a distance of 1 from the center.

3. What is exactness in relation to differential forms?

Exactness is a property of a differential form that means it can be written as the exterior derivative of another form. In other words, an exact form is the derivative of some other form. This property is important in calculus and differential geometry, as it allows for integration of exact forms over a manifold.

4. How is the differential form homework on unit sphere in R3 solved?

The differential form homework on unit sphere in R3 is solved by using the properties of exactness and the unit sphere in R3. First, the differential form is checked for exactness. If it is exact, then the homework involves finding the form that it is the derivative of, and using this to calculate integrals over the unit sphere.

5. What are the real-life applications of differential forms on unit sphere in R3?

Differential forms on unit sphere in R3 have various real-life applications, such as in physics, engineering, and computer graphics. They are used to describe physical quantities, such as electric and magnetic fields, and to solve problems involving curved surfaces, such as in fluid dynamics. They are also used in computer graphics to model and render 3D surfaces and objects.

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