# Solve Integral on Curve C of Sphere

• imana41
In summary, the conversation discusses how to solve for the largest circle on a sphere, with one person suggesting using integrals on a curve in higher-dimensional space and providing an example solution. Another person suggests a different approach using a great circle parallel to the xy-plane and provides the necessary equations. The conversation also discusses the infinite number of possible great circles and their symmetry. Ultimately, the integral over the curve results in 0.
imana41
solve
on curve c that is the bigest circle of sphere [URL]http://latex.codecogs.com/gif.latex?(x-1)^2+(y-1)^2+(z-1)^2=1[/URL]

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What have you tried until now? Do you know how to do integrals on a curve in a higher-dimensional space (like, for example, integrals on a curve in the plane)?

i know F=(M,N,P)=(yz,xz,xy) and c is on closed curve i think i should use
but the euqals is zero I'm not sure . plaese help

This cannot be since dσ is two-dimensional, while dx dy dz is three-dimensional, and your integral is a line integral, i.e. one-dimensional. So no.

If you have an integral over a curve (a line integral), this is per definition
$$\int_C f_1(x, y, ...) \mathrm{d} x + f_2(x, y, ...) \mathrm{d} y + ... = \int_C f_1(x(t), y(t), ...) \frac{\partial x(t)}{\partial t} \mathrm{d} t + f_2(x(t), y(t), ...) \frac{\partial y(t)}{\partial t} \mathrm{d} t + ...$$, where x(t), y(t), ... describe your curve. Can you give me x(t), y(t) and z(t) for your curve?

if [URL]http://latex.codecogs.com/gif.latex?x^2+y^2+z^2=1[/URL] we get x(t)=cost , y(t)=sint and z(t)=1 but for [URL]http://latex.codecogs.com/gif.latex?(x-1)^2+(y-1)^2+(z-1)^2=1[/URL] I'm not know ??

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What if you try
$$x(t) = 1+cos(t) , y(t) = 1+sin(t) , z(t) = 1$$
Then you get
$$(1 + cos(t) -1)^{2} + (1 + sin(t) - 1)^{2} + (1-1)^{2} = 1$$
thus
$$cos^{2}(t) + sin^{2}(t) = 1$$

imana41 said:
if [URL]http://latex.codecogs.com/gif.latex?x^2+y^2+z^2=1[/URL] we get x(t)=cost , y(t)=sint and z(t)=1 but for [URL]http://latex.codecogs.com/gif.latex?(x-1)^2+(y-1)^2+(z-1)^2=1[/URL] I'm not know ??

Sure you meant z = 0 there. Just as Clever-Name has said, adding 1 to each coordinate then gives you the curve for your problem. (If it is your curve, I'm not quite sure what a "bigest circle of sphere" is.).

Then just plug everything into the integral, you should get something like
$$\int_{0}^{2 \pi} \left[ - \sin t - \sin^2 t + \cos t + \cos^2 t \right] \mathrm{d} t = 0$$.

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The "biggest circle on a sphere" is simply as "great circle" and there are an infinite number of them. The simplest thing to do is to take the great circle that is parallel to the xy-plane, given by x= 1+ cos(t), y= 1+ sin(t), z= 1.
Then dx= -sin(t)dt, dy= cos(t)dt, and dz= 0. yz= 1+ sin(t), xz= 1+ cos(t), and xy= 1+ sin(t)+ cos(t)+ sin(t)cos(t) but since that will be multiplied by dz= 0, it doesn't matter. The integral will be
$$\int_{t=0}^{2\pi} [-(1+ sin(t))sin(t)+ (1+ cos(t))cos(t)]dt$$
The identity $cos^2(t)- sin^2(t)= cos(2t)$ might be helpful.

I said there are an infinite number of great circles but it should be evident from the symmetry that they all give the same integral.

HallsofIvy said:
The "biggest circle on a sphere" is simply as "great circle" and there are an infinite number of them. The simplest thing to do is to take the great circle that is parallel to the xy-plane, given by x= 1+ cos(t), y= 1+ sin(t), z= 1.
Then dx= -sin(t)dt, dy= cos(t)dt, and dz= 0. yz= 1+ sin(t), xz= 1+ cos(t), and xy= 1+ sin(t)+ cos(t)+ sin(t)cos(t) but since that will be multiplied by dz= 0, it doesn't matter. The integral will be
$$\int_{t=0}^{2\pi} [-(1+ sin(t))sin(t)+ (1+ cos(t))cos(t)]dt$$
The identity $cos^2(t)- sin^2(t)= cos(2t)$ might be helpful.

I said there are an infinite number of great circles but it should be evident from the symmetry that they all give the same integral.

## What is an integral on a curve of sphere?

An integral on a curve of sphere is a mathematical concept that calculates the area under a curve on the surface of a sphere. It is similar to finding the area under a curve on a flat surface, but takes into account the curve's position on the curved surface of a sphere.

## How is an integral on a curve of sphere solved?

An integral on a curve of sphere is solved using a process called parametrization, where the curve is broken down into smaller segments and each segment is represented by a set of coordinates. These coordinates are then used in the integral formula to find the total area under the curve.

## What are the applications of solving an integral on a curve of sphere?

Solving an integral on a curve of sphere has many real-world applications, such as calculating the surface area of a spherical object, finding the volume of a solid with a curved surface, and predicting the path of a moving object on a spherical surface.

## What is the difference between an integral on a curve of sphere and a regular integral?

The main difference between an integral on a curve of sphere and a regular integral is that the former takes into account the curvature of the surface, while the latter assumes a flat surface. This makes the process of solving the integral on a curve of sphere more complex and requires a different approach.

## Are there any limitations to solving an integral on a curve of sphere?

Yes, there are some limitations to solving an integral on a curve of sphere. It is only applicable to curves on a spherical surface and cannot be used for other types of curved surfaces. Additionally, the parametrization process can become more complex for more intricate curves, making it difficult to find an exact solution.

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