symmetry

Examples of symmetry in the following topics:

• Cylindrical and Spherical Coordinates

  • 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.
  • Cylindrical coordinates are useful in connection with objects and phenomena that have some rotational symmetry about the longitudinal axis, such as water flow in a straight pipe with a round cross-section, heat distribution in a metal cylinder, electromagnetic fields produced by an electric current in a long, straight wire, and so on.
  • Spherical coordinates are useful in connection with objects and phenomena that have spherical symmetry, such as an electric charge located at the origin.

• Double Integrals in Polar Coordinates

  • 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.
  • This is the case because the function has a cylindrical symmetry.
  • In general, the best practice is to use the coordinates that match the built-in symmetry of the function.

• Triple Integrals in Spherical Coordinates

  • When the function to be integrated has a spherical symmetry, change the variables into spherical coordinates and then perform integration.
  • When the function to be integrated has a spherical symmetry, it is sensible to change the variables into spherical coordinates and then perform integration.
  • In $R^3$ some domains have a spherical symmetry, so it's possible to specify the coordinates of every point of the integration region by two angles and one distance.
  • Spherical coordinates are useful when domains in $R^3$ have spherical symmetry.

• Curve Sketching

  • Determine the symmetry of the curve.
  • If the exponent of $x$ is always even in the equation of the curve, then the $y$-axis is an axis of symmetry for the curve.
  • Similarly, if the exponent of $y$ is always even in the equation of the curve, then the $x$-axis is an axis of symmetry for the curve.

• Triple Integrals in Cylindrical Coordinates

  • 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.

• Applications to Economics and Biology

  • Since we can assume that there is a cylindrical symmetry in the blood vessel, we first consider the volume of blood passing through a ring with inner radius $r$ and outer radius $r+dr$ per unit time ($dF$):

• Linear and Quadratic Functions

  • The graph of a quadratic function is a parabola whose axis of symmetry is parallel to the y-axis .