coordinates

Examples of coordinates in the following topics:

• Introduction to Spherical and Cylindrical Harmonics

  • In this section we will apply separation of variables to Laplace's equation in spherical and cylindrical coordinates.
  • 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.
  • The disadvantage to using coordinate systems other than Cartesian is that the differential operators are more complicated.

• Unit Vectors and Multiplication by a Scalar

  • The unit vectors are different for different coordinates.
  • In Cartesian coordinates the directions are x and y usually denoted $\hat{x}$ and $\hat{y}$.
  • The unit vectors in Cartesian coordinates describe a circle known as the "unit circle" which has radius one.
  • This can be seen by taking all the possible vectors of length one at all the possible angles in this coordinate system and placing them on the coordinates.

• Four-Dimensional Space-Time

  • Observer A sets up a space-time coordinate system (t, x, y, z); similarly, A' sets up his own space-time coordinate system (t', x', y', z').
  • You should not find it odd to work with four dimensions; any time you have to meet your friend somewhere you have to tell him four variables: where (three spatial coordinates) and when (one time coordinate).
  • where $(T, X, Y, Z)$ refers to the coordinates in either frame.
  • For the coordinate transformation in , the transformations are:
  • Two coordinate systems in which the primed frame moves with velocity v with respect to the unprimed frame

• Components of a Vector

  • Vectors are geometric representations of magnitude and direction which are often represented by straight arrows, starting at one point on a coordinate axis and ending at a different point.
  • A vector is defined by its magnitude and its orientation with respect to a set of coordinates.
  • To visualize the process of decomposing a vector into its components, begin by drawing the vector from the origin of a set of coordinates.
  • The horizontal component stretches from the start of the vector to its furthest x-coordinate.

• Adding and Subtracting Vectors Graphically

  • The head-to-tail method of vector addition requires that you lay out the first vector along a set of coordinate axes.
  • To start, draw a set of coordinate axes.
  • Next, draw out the first vector with its tail (base) at the origin of the coordinate axes.
  • The head-to-tail method of vector addition requires that you lay out the first vector along a set of coordinate axes.

• Lorentz Transformations

  • Let's also assume that their coordinate systems coincide at $ t = 0$, and that one emits a light pulse at $t = t' = 0$ from $x = x' = 0$.

• Stress and Strain

  • The above equation is defined in radial coordinates which can be seen in .
  • The electric field of a point charge is defined in radial coordinates.

• Electric Field from a Point Charge

  • The above equation is defined in radial coordinates, which can be seen in .
  • The electric field of a point charge is defined in radial coordinates.

• Scalars vs. Vectors

  • The two parts are its length which represents the magnitude and its direction with respect to some set of coordinate axes.
  • Typically this reference point is a set of coordinate axes like the x-y plane.

• Reference Frames and Displacement

  • This is referred to as choosing a coordinate system, or choosing a frame of reference.
  • But you don't want to change coordinate systems in the middle of a calculation.
  • But this shows that there is a third arbitrary choice that goes into choosing a coordinate system: valid frames of reference can differ from each other by moving relative to one another.
  • It might seem strange to use a coordinate system moving relative to the earth -- but, for instance, the frame of reference moving along with a train might be far more convenient for describing things happening inside the train.