Line integral in polar/spherical system?

In summary, the conversation discusses the possibility of defining a vector field and line integrals in polar and spherical coordinate systems. The calculation for the differential of arc length in polar coordinates is shown, as well as the formula for the differential of arc length in spherical coordinates. The equations for the differentials of x, y, and z in terms of d\rho, d\theta, and d\phi are also provided.
  • #1
player1_1_1
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Hello, sorry for my English;D

Homework Statement


Can a vector field exist in polar/spherical system? is it possible to define line integral in these systems? does it make any sense a vector field defined in polar system, ex. [tex]\vec A\left(r,\varphi\right)=r^3[/tex]? and a line integral from [tex]r_1,\varphi_1[/tex] to [tex]r_2,\varphi_2[/tex] like this [tex]\int\limits_L\vec A\left(r,\varphi\right)\mbox{d}r+r\vec A\left(r,\varphi\right)\mbox{d}\varphi[/tex], where L is a line defined by [tex]r\left(\phi\right)[/tex] equation or [tex]r=r(t),\quad\phi=\phi(t)[/tex]
and for spherical system [tex]\int\limits_L\vec A\left(r,\varphi,\phi\right)\mbox{d}r+r\vec A\left(r,\varphi,\phi\right)\mbox{d}\varphi+r\vec A\left(r,\varphi,\phi\right)\mbox{d}\phi[/tex]?
thanks!
 
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  • #2
In polar coordinates, [itex]x= r cos(\phi)[/itex] so [itex]dx= cos(\phi)dr- r sin(\phi)d\phi[/itex]. [itex]y= r sin(\phi)[/itex] so [itex]dy= sin(\phi)dr+ r cos(\phi)d\phi[/itex].

[itex]dx^2= cos^2(\phi)dr^2- 2r cos(\phi)sin(\phi)drd\phi+ r^2 sin^2(\phi)d\phi^2[/itex]
[itex]dy^2= sin^2(\theat)dr^2+ 2r cos(\phi)sin(\phi)drd\phi+ r^2cos^2(\phi)d\phi^2[/itex]

so [itex]dx^2+ dy^2= dr^2+ r^2 d\phi^2[/itex] so the "differential of arc length", ds, is given by [itex]ds= \sqrt{dr^2+ r^2d\phi^2}[/itex].

If r and [itex]\phi[/itex] are given in terms of a parameter, t, then
[itex]ds= \sqrt{\left(\frac{dr}{dt}\right)^2+ r^2\left(\frac{d\phi}{dt}\right)^2}dt[/itex]

In spherical coordinates:

Since [itex]x= \rho cos(\theta)sin(\phi)[/itex], [itex]dx= cos(\theta)sin(\phi)d\rho- \rho sin(\theta)sin(\phi)d\phi+ \rho cos(\theta)cos(\phi)d\phi[/itex].

Since [itex]y= \rho sin(\theta)sin(\phi)[/itex], [itex]dx= sin(\theta)sin(\phi)d\rho+ \rho cos(\theta)sin(\phi)d\phi+ \rho sin(\theta)cos(\phi)d\phi[/itex].

Since [itex]z= \rho cos(\phi)[/itex], [itex]dz= cos(\phi)d\rho- \rho sin(\phi)d\phi[/itex].

The "differential of areclength", ds, is given by [itex]ds= \sqrt{dx^2+ dy^2+ dz^2}[/itex]. Use the above equations to write that in terms of [itex]d\rho, d\theta[/itex], and [itex]d\phi[/itex]. it starts out messy but there is a lot of cancelling at the end.
 
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  • #3
Yes, a vector field can exist in polar/spherical systems. In fact, many physical systems are modeled using polar or spherical coordinates. It is possible to define a line integral in these systems, and it makes sense to do so. The vector field \vec A\left(r,\varphi\right)=r^3 is a valid vector field in polar coordinates, and the line integral from r_1,\varphi_1 to r_2,\varphi_2 as described is a valid way to calculate the work done by this vector field along the given path. Similarly, for spherical coordinates, the line integral \int\limits_L\vec A\left(r,\varphi,\phi\right)\mbox{d}r+r\vec A\left(r,\varphi,\phi\right)\mbox{d}\varphi+r\vec A\left(r,\varphi,\phi\right)\mbox{d}\phi is a valid way to calculate the work done by the vector field along the given path. So, in summary, line integrals can be defined and make sense in polar and spherical systems, and they are useful tools for calculating work done by vector fields in these coordinate systems.
 

Related to Line integral in polar/spherical system?

What is a line integral in polar/spherical system?

A line integral in polar/spherical system is a type of integral used to calculate the total work done by a vector field along a curve in polar or spherical coordinates. It takes into account both the magnitude and direction of the vector field along the curve.

How is a line integral in polar/spherical system calculated?

To calculate a line integral in polar/spherical system, the curve must first be parameterized in terms of the polar/spherical coordinates. Then, the integral is evaluated using the appropriate formula based on the coordinates and the vector field. This typically involves converting the vector field into polar/spherical coordinates and then integrating along the curve.

What is the significance of line integrals in polar/spherical system?

Line integrals in polar/spherical system are important in many areas of science and engineering, particularly in electromagnetism and fluid mechanics. They allow us to calculate the work done by a vector field along a curve, which can help us understand the behavior of the field and its impact on the surrounding environment.

What are some real-world applications of line integrals in polar/spherical system?

Line integrals in polar/spherical system have many practical applications. For example, they are used in physics to calculate the electric and magnetic fields generated by charged particles or moving currents. They are also used in engineering to analyze the flow of fluids in pipes or around objects, and in computer graphics to render 3D images using spherical coordinates.

Are there any limitations to using line integrals in polar/spherical system?

While line integrals in polar/spherical system are a powerful tool in many applications, they do have some limitations. One limitation is that they are only valid for conservative vector fields, which means that the work done along a closed curve must be independent of the path taken. Additionally, they can be difficult to calculate for complex curves or vector fields with changing directions.

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