which course polar coordinates

Which set of polar coordinates will be the same as our first?

Video Description: Herb Gross defines and demonstrates the use of polar coordinates. He describes the non-uniqueness of polar coordinates and how to calculate the slope of a curve, which depends on the angle the curve makes with the radius vector. Finally, he computes the area (in terms of polar coordinates) of the region between two rays.

What is an example of a polar coordinate?

POLAR COORDINATES AND CELESTIAL MECHANICS In class, we showed that the acceleration vector in plane polar (r, f) coordinates can be written as : a = r (1)..-rf °2 r ` + r f.. +2r°f ° f ` where r is the distance from the origin and f is the azimuthal angle.

What is the range of the polar coordinate system?

Like Cartesian coordinates, polar coordinates are used to identify the locations of points in the plane. The polar coordinate system will be useful for many problems you encounter at MIT, such as those involving circular motion or radial forces. Lecture Video and Notes Video Excerpts. Clip 1: Introduction to Polar Coordinates

What are 3D polar coordinates?

The polar coordinates are [latex](3\sqrt2,\frac{\pi}{4})[/latex]. Note that [latex]r^2 = 18[/latex] implies [latex]r=\pm\sqrt{18}[/latex]. We chose to ignore the negative [latex]r[/latex] value. Also note that [latex]\tan^{-1}\left( 1 \right)[/latex] has many answers. This corresponds to the non-uniqueness of polar coordinates.

What is the polar coordinate?

Polar coordinates are points labeled (r,θ) ( r, θ) and plotted on a polar grid. The polar grid is represented as a series of concentric circles radiating out from the pole, or the origin of the coordinate plane. The reference point (analogous to the origin of a Cartesian system) is called the pole, and the ray from the pole in ...

How is polar coordinate determined?

In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction.

What are the coordinates of a rectangular grid?

The rectangular coordinates are (0,3) ( 0, 3). Polar and Coordinate Grid of Equivalent Points: The rectangular coordinate (0,3) ( 0, 3) is the same as the polar coordinate (3,π 2) ( 3, π 2) as plotted on the two grids above.

What does a positive angular coordinate mean?

In many contexts, a positive angular coordinate means that the angle ϕ ϕ is measured counterclockwise from the axis. In mathematical literature, the polar axis is often drawn horizontal and pointing to the right.

What is the reference point of a Cartesian system?

The reference point (analogous to the origin of a Cartesian system) is called the pole, and the ray from the pole in the reference direction is the polar axis. The distance from the pole is called the radial coordinate or radius, and the angle is called the angular coordinate, polar angle, or azimuth. The radial coordinate is often denoted by r r ...

What is the origin of polar coordinates?

In polar coordinates the origin is often called the pole. Because we aren’t actually moving away from the origin/pole we know that r = 0 r = 0. However, we can still rotate around the system by any angle we want and so the coordinates of the origin/pole are (0,θ) ( 0, θ).

What is the difference between polar and cartesian coordinates?

In Cartesian coordinates there is exactly one set of coordinates for any given point. With polar coordinates this isn’t true. In polar coordinates there is literally an infinite number of coordinates for a given point.

Plotting Points Using Polar Coordinates

When we think about plotting points in the plane, we usually think of rectangular coordinates (x,y) ( x, y) in the Cartesian coordinate plane. However, there are other ways of writing a coordinate pair and other types of grid systems.

Converting Between Polar Coordinates to Rectangular Coordinates

When given a set of polar coordinates, we may need to convert them to rectangular coordinates. To do so, we can recall the relationships that exist among the variables x,y,r x, y, r, and θ θ .

Converting from Rectangular Coordinates to Polar Coordinates

To convert rectangular coordinates to polar coordinates, we will use two other familiar relationships. With this conversion, however, we need to be aware that a set of rectangular coordinates will yield more than one polar point.

Transforming Equations between Polar and Rectangular Forms

We can now convert coordinates between polar and rectangular form. Converting equations can be more difficult, but it can be beneficial to be able to convert between the two forms.

Identify and Graph Polar Equations by Converting to Rectangular Equations

We have learned how to convert rectangular coordinates to polar coordinates, and we have seen that the points are indeed the same. We have also transformed polar equations to rectangular equations and vice versa. Now we will demonstrate that their graphs, while drawn on different grids, are identical.

Key Concepts

The polar grid is represented as a series of concentric circles radiating out from the pole, or origin.

What are some examples of polar coordinates?

Polar Coordinates Examples. Example 1: Convert the polar coordinate (4, π/2) to a rectangular point. Solution: Given, We know that, Hence, the rectangular coordinate of the point is (0, 4). Example 2: Convert the rectangular or cartesian coordinates (2 , 2) to polar coordinates.

What is the polar axis?

Pole = The reference point. Polar axis = the line segment ray from the pole in the reference direction. In the polar coordinate system, the origin is called a pole. Here, instead of representing the point as (x, y), we can express it as a polar coordinate (r, θ). Where the value of r can be negative.

What is the name of the coordinate system where each point is decided by a distance from a reference point?

Polar Coordinate System. When each point on a plane of a two-dimensional coordinate system is decided by a distance from a reference point and an angle is taken from a reference direction, it is known as the polar coordinate system .

What is the distance of a point from the y axis called?

In the Cartesian coordinate system, the distance of a point from the y-axis is called its x-coordinate and the distance of a point from the x-axis is called its y -coordinate.