normal

Examples of normal in the following topics:

• Tangent Vectors and Normal Vectors

  • An object is normal to another object if it is perpendicular to the point of reference.
  • Not only can vectors be ‘normal' to objects, but planes can also be normal.
  • Tangent vectors are almost exactly like normal vectors, except they are tangent instead of normal to the other vector or object.
  • This plane is normal to the point on the sphere to which it is tangent.
  • Each point on the sphere will have a unique normal plane.

• Physics and Engineering: Fluid Pressure and Force

  • Mathematically, $p = \frac{F}{A}$, where $p$ is the pressure, $\mathbf{F}$ is the normal force, and $A$ is the area of the surface on contact.
  • It relates the vector surface element (a vector normal to the surface) with the normal force acting on it.
  • The pressure is the scalar proportionality constant that relates the two normal vectors:
  • The subtraction (–) sign comes from the fact that the force is considered towards the surface element while the normal vector points outward.
  • The total force normal to the contact surface would be:

• Surface Integrals of Vector Fields

  • Alternatively, if we integrate the normal component of the vector field, the result is a scalar.
  • This also implies that if $\mathbf{v}$ does not just flow along $S$—that is, if $\mathbf{v}$ has both a tangential and a normal component—then only the normal component contributes to the flux.
  • Based on this reasoning, to find the flux, we need to take the dot product of $\mathbf{v}$ with the unit surface normal to $S$, at each point, which will give us a scalar field, and integrate the obtained field as above.
  • The cross product on the right-hand side of this expression is a surface normal determined by the parametrization.
  • An illustration of the Kelvin–Stokes theorem, with surface $\Sigma$, its boundary $\partial$, and the "normal" vector $n$.

• Double Integrals Over General Regions

  • $x$-axis: If the domain $D$ is normal with respect to the $x$-axis, and $f:D \to R$ is a continuous function, then $\alpha(x)$  and $\beta(x)$ (defined on the interval $[a, b]$) are the two functions that determine $D$.
  • $y$-axis: If $D$ is normal with respect to the $y$-axis and $f:D \to R$ is a continuous function, then $\alpha(y)$ and $\beta(y)$ (defined on the interval $[a, b]$) are the two functions that determine $D$.
  • This domain is normal with respect to both the $x$- and $y$-axes.
  • In this case the two functions are $\alpha (x) = x^2$ and $\beta (x) = 1$, while the interval is given by the intersections of the functions with $x=0$, so the interval is $[a,b] = [0,1]$ (normality has been chosen with respect to the $x$-axis for a better visual understanding).
  • Double integral over the normal region $D$ shown in the example.

• Vectors in the Plane

  • which we call the point-normal equation of the plane and is the general equation we use to describe the plane.
  • Calculate the directions of the normal vector and the directional vector of a reference point

• Probability

  • Probability distribution function of a normal (or Gaussian) distribution, where mean $\mu=0$  and variance $\sigma^2=1$.

• Triple Integrals

  • If $T$ is a domain that is normal with respect to the xy-plane and determined by the functions $\alpha (x,y)$ and $\beta(x,y)$, then:
  • Example of domain in $R^3$ that is normal with respect to the $xy$-plane.

• The Cross Product

  • Because it is perpendicular to both original vectors, the resulting vector is normal to the plane of the original vectors.

• Change of Variables

  • The limits of integration are often not easily interchangeable (without normality or with complex formulae to integrate).

• Parametric Surfaces and Surface Integrals

  • An illustration of the Kelvin–Stokes theorem, with surface $\Sigma$, its boundary $\partial$, and the "normal" vector $\mathbf{n}$.