map projection

Examples of map projection in the following topics:

• Essential Functions for Mathematical Modeling

  • A geographical map projection of a region of the earth onto a small, plane surface is a model which can be used for many purposes such as planning travel.

• Inverse Functions

  • If $f$ maps $X$ to $Y$, then $f^{-1}$  maps $Y$ back to $X$.
  • Because $f$ maps $a$ to $3$, the inverse $f^{-1}$ maps $3$ back to $a$.

• Conic Sections

  • In projective geometry, the conic sections in the projective plane are equivalent to each other up to projective transformations.

• Inverse Functions

  • Because $f$ maps $a$ to 3, the inverse $f^{-1}$ maps 3 back to $a$.

• Real Numbers, Functions, and Graphs

  • Here, the domain is the entire set of real numbers and the function maps each real number to its square.

• Intermediate Value Theorem

  • The intermediate value theorem states that for each value between the least upper bound and greatest lower bound of the image of a continuous function there is at least one point in its domain that the function maps to that value.

• Three-Dimensional Coordinate Systems

  • $\varphi$: the angle from the reference direction to the orthogonal plane projected by the directional vector of $r$

• Curl and Divergence

  • The curl of a vector field $\mathbf{F}$, denoted by $\nabla \times \mathbf{F}$, is defined at a point in terms of its projection onto various lines through the point.
  • If $\hat{\mathbf{n}}$ is any unit vector, the projection of the curl of $\mathbf{F}$ onto $\hat{\mathbf{n}}$ is defined to be the limiting value of a closed line integral in a plane orthogonal to $\hat{\mathbf{n}}$ as the path used in the integral becomes infinitesimally close to the point, divided by the area enclosed.

• Cylindrical and Spherical Coordinates

  • 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.

• Directional Derivatives and the Gradient Vector

  • The gradient of the function $f(x,y) = −\left((\cos x)^2 + (\cos y)^2\right)$ depicted as a projected vector field on the bottom plane.