Examples of polar coordinate in the following topics:
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- 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 $(−\pi, \pi]$:
- Particularly in this case, you can see that the representation of the function f became simpler in polar coordinates.
- This figure illustrates graphically a transformation from cartesian to polar coordinates
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- When given a set of polar coordinates, we may need to convert them to rectangular coordinates.
- To convert rectangular coordinates to polar coordinates, we will use two other familiar relationships.
- This corresponds to the non-uniqueness of polar coordinates.
- A right triangle with rectangular (Cartesian) coordinates and equivalent polar coordinates.
- A right triangle with rectangular (Cartesian) coordinates and equivalent polar coordinates.
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- Polar coordinates define the location of an object in a plane by using a distance and an angle from a reference point and axis.
- Such definitions are called polar coordinates.
- Polar coordinates in $r$ and $\theta$ can be converted to Cartesian coordinates $x$ and $y$.
- A set of polar coordinates.
- Use a polar coordinate to define a point with $r$ (distance from pole), and $\theta$(angle between axis and ray)
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- Conic sections are sections of cones and can be represented by polar coordinates.
- In polar coordinates, a conic section with one focus at the origin is given by the following equation:
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- The polar coordinate system is an alternate coordinate system where the two variables are $r$ and $\theta$, instead of $x$ and $y$.
- Polar coordinates are points labeled $(r,θ)$ and plotted on a polar grid.
- The distance from the pole is called the radial coordinate or radius, and the angle is called the angular coordinate, polar angle, or azimuth.
- Even though we measure
$θ$ first and then $r$, the polar point is written with the $r$
-coordinate first.
- Points in the polar coordinate system with pole $0$ and polar axis $L$.
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- Area and arc length are calculated in polar coordinates by means of integration.
- Since it can be very difficult to measure the length of an arc linearly, the solution is to use polar coordinates.
- Using polar coordinates allows us to integrate along the length of the arc in order to compute its length.
- To find the area enclosed by the arcs and the radius and polar angles, you again use integration.
- Evaluate arc segment area and arc length using polar coordinates and integration
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- Some curves have a simple expression in polar coordinates, whereas they would be very complex to represent in Cartesian coordinates.
- To graph in the rectangular coordinate system we construct a table of $x$ and $y$ values.
- To graph in the polar coordinate system we construct a table of $r$ and $\theta$ values.
- We enter values of $\theta$ into a polar equation and calculate $r$.
- Polar equations can be used to generate unique graphs.
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- Polar coordinates allow conic sections to be expressed in an elegant way.
- We can define any conic in the polar coordinate system in terms of a fixed point, the focus $P(r,θ)$ at the pole, and a line, the directrix, which is perpendicular to the polar axis.
- Thus, each conic may be written as a polar equation in terms of $r$ and $\theta$.
- For a conic with a focus at the origin, if the directrix is $x=±p$, where $p$ is a positive real number, and the eccentricity is a positive real number $e$, the conic has a polar equation:
- For a conic with a focus at the origin, if the directrix is $y=±p$, where $p$ is a positive real number, and the eccentricity is a positive real number $e$, the conic has a polar equation:
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- Complex numbers can be represented in polar coordinates using the formula $a+bi=re^{i\theta}$.
- The previous geometric idea where the number $z=a+bi$ is associated with the point $(a,b)$ on the usual $xy$-coordinate system is called rectangular coordinates.
- The alternative way to picture things is called polar coordinates.
- In polar coordinates, the parameters are $r$ and $\phi$.
- Explain how to represent complex numbers in polar coordinates and why it is useful to do so
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- While Cartesian coordinates have many applications, cylindrical and spherical coordinates are useful when describing objects or phenomena with specific symmetries.
- 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 spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the radial distance of that point from a fixed origin, its polar angle measured from a fixed zenith direction, and the azimuth angle of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the zenith, measured from a fixed reference direction on that plane.
- Spherical coordinates ($r$, $\theta$, $\varphi$) as often used in mathematics: radial distance $r$, azimuthal angle $\theta$, and polar angle $\varphi$.
- A cylindrical coordinate system with origin $O$, polar axis $A$, and longitudinal axis $L$.