Examples of cylindrical coordinate in the following topics:
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- When the function to be integrated has a cylindrical symmetry, it is sensible to integrate using cylindrical coordinates.
- When the function to be integrated has a cylindrical symmetry, it is sensible to change the variables into cylindrical coordinates and then perform integration.
- Also in switching to cylindrical coordinates, the $dx\, dy\, dz$ differentials in the integral become $\rho \, d\rho \,d\varphi \,dz$.
- Finally, it is possible to apply the final formula to cylindrical coordinates:
- Cylindrical coordinates are often used for integrations on domains with a circular base.
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- Cylindrical and spherical coordinates are useful when describing objects or phenomena with specific symmetries.
- While Cartesian coordinates have many applications, cylindrical and spherical coordinates are useful when describing objects or phenomena with specific symmetries.
- For the conversion between cylindrical and Cartesian coordinate co-ordinates, it is convenient to assume that the reference plane of the former is the Cartesian $xy$-plane (with equation $z = 0$), and the cylindrical axis is the Cartesian $z$-axis.
- Then the $z$ coordinate is the same in both systems, and the correspondence between cylindrical $(\rho,\varphi)$ and Cartesian $(x,y)$ are the same as for polar coordinates, namely $x = \rho \cos \varphi; \, y = \rho \sin \varphi$.
- A cylindrical coordinate system with origin $O$, polar axis $A$, and longitudinal axis $L$.
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- The three-dimensional coordinate system expresses a point in space with three parameters, often length, width and depth ($x$, $y$, and $z$).
- The cylindrical system uses two linear parameters and one radial parameter:
- This is a three dimensional space represented by a Cartesian coordinate system.
- The cylindrical coordinate system is like a mix between the spherical and Cartesian system, incorporating linear and radial parameters.
- Identify the number of parameters necessary to express a point in the three-dimensional coordinate system
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- To do so, the function must be adapted to the new coordinates.
- Changing to cylindrical coordinates may be useful depending on the setup of problem.
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- In this section we will apply separation of variables to Laplace's equation in spherical and cylindrical coordinates.
- Laplace's equation is important in its own right as the cornerstone of potential theory, but the wave equation also involves the Laplacian derivative, so the ideas discussed in this section will be used to build solutions of the wave equation in spherical and cylindrical coordinates too.
- Spherical coordinates are important when treating problems with spherical or nearly-spherical symmetry.
- For instance, in Cartesian coordinates the surface of the unit cube can be represented by:
- On the other hand, if we tried to use Cartesian coordinates to solve a boundary value problem on a spherical domain, we couldn't represent this as a fixed value of any of the coordinates.
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- When domain has a cylindrical symmetry and the function has several specific characteristics, apply the transformation to polar coordinates.
- In $R^2$, if the domain has a cylindrical symmetry and the function has several particular characteristics, you can apply the transformation to polar coordinates, which means that the generic points $P(x, y)$ in Cartesian coordinates switch to their respective points in polar coordinates.
- The polar coordinates $r$ and $\varphi$ can be converted to the Cartesian coordinates $x$ and $y$ by using the trigonometric functions sine and cosine:
- The Cartesian coordinates $x$ and $y$ can be converted to polar coordinates $r$ and $\varphi$ with $r \geq 0$ and $\varphi$ in the interval $(−\pi, \pi]$:
- This is the case because the function has a cylindrical symmetry.
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- When the function to be integrated has a spherical symmetry, change the variables into spherical coordinates and then perform integration.
- It's possible to use therefore the passage in spherical coordinates; the function is transformed by this relation:
- Points on $z$-axis do not have a precise characterization in spherical coordinates, so $\theta$ can vary from $0$ to $2 \pi$.
- Finally, you obtain the final integration formula: It's better to use this method in case of spherical domains and in case of functions that can be easily simplified, by the first fundamental relation of trigonometry, extended in $R^3$; in other cases it can be better to use cylindrical coordinates.
- Spherical coordinates are useful when domains in $R^3$ have spherical symmetry.
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- or that isentropic stars must have constant angular velocity on cylindrical surfaces.
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- In the shell method, a function is rotated around an axis and modeled by an infinite number of cylindrical shells, all infinitely thin.
- In the integrand, the factor $x$ represents the radius of the cylindrical shell under consideration, while is equal to the height of the shell.
- Therefore, the entire integrand, $2\pi x \left | f(x) - g(x) \right | \,dx$, is nothing but the volume of the cylindrical shell.
- Each segment located at $x$, between $f(x)$and the $x$-axis, gives a cylindrical shell after revolution around the vertical axis.
- Use shell integration to create a cylindrical shell and calculate the volume of a "solid of revolution" perpendicular to the axis of revolution.
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- These muscle fibers are cylindrical, multinucleate in nature.
- The involuntary contraction of cardiac muscle is coordinated by the intercalated disks, so the entire heart beats in a controlled, uniform manner, ensuring that blood is efficiently pumped from the chambers.