scalar function

Examples of scalar function in the following topics:

• Line Integrals

  • A line integral is an integral where the function to be integrated is evaluated along a curve.
  • A line integral (sometimes called a path integral, contour integral, or curve integral) is an integral where the function to be integrated is evaluated along a curve.
  • The function to be integrated may be a scalar field or a vector field.
  • The value of the line integral is the sum of the values of the field at all points on the curve, weighted by some scalar function on the curve (commonly arc length or, for a vector field, the scalar product of the vector field with a differential vector in the curve).
  • For some scalar field $f:U \subseteq R^n \to R$, the line integral along a piecewise smooth curve $C \subset U$ is defined as:

• Limits and Continuity

  • A study of limits and continuity in multivariable calculus yields counter-intuitive results not demonstrated by single-variable functions.
  • For example, there are scalar functions of two variables with points in their domain which give a particular limit when approached along any arbitrary line, yet give a different limit when approached along a parabola.
  • For example, the function $f(x,y) = \frac{x^2y}{x^4+y^2}$ approaches zero along any line through the origin.
  • Continuity in single-variable function as shown is rather obvious.
  • However, continuity in multivariable functions yields many counter-intuitive results.

• Parametric Surfaces and Surface Integrals

  • The simplest type of parametric surfaces is given by the graphs of functions of two variables: $z=f(x,y)$; $\vec{r}(x,y)=(x,y,f(x,y))$.
  • Given a surface, one may integrate over its scalar fields (that is, functions which return scalars as values), and vector fields (that is, functions which return vectors as values).

• Conservative Vector Fields

  • A conservative vector field is a vector field which is the gradient of a function, known in this context as a scalar potential.
  • A conservative vector field is a vector field which is the gradient of a function, known in this context as a scalar potential.
  • Definition: A vector field $\mathbf{v}$ is said to be conservative if there exists a scalar field $\varphi$ such that $\mathbf{v}=\nabla\varphi$.
  • When the above equation holds, $\varphi$ is called a scalar potential for $\mathbf{v}$.
  • For any scalar field $\varphi$: $\nabla \times \nabla \varphi=0$.

• Functions of Several Variables

  • As we will see, multivariable functions may yield counter-intuitive results when applied to limits and continuity.
  • Unlike a single variable function $f(x)$, for which the limits and continuity of the function need to be checked as $x$ varies on a line ($x$-axis), multivariable functions have infinite number of paths approaching a single point.Likewise, the path taken to evaluate a derivative or integral should always be specified when multivariable functions are involved.
  • We have also studied theorems linking derivatives and integrals of single variable functions.
  • A scalar field shown as a function of $(x,y)$.
  • Extensions of concepts used for single variable functions may require caution.

• Calculus of Vector-Valued Functions

  • A vector function is a function that can behave as a group of individual vectors and can perform differential and integral operations.
  • A vector-valued function, also referred to as a vector function, is a mathematical function of one or more variables whose range is a set of multidimensional vectors or infinite-dimensional vectors.
  • The input of a vector-valued function could be a scalar or a vector.
  • When taking the derivative of a vector function, the function should be treated as a group of individual functions.
  • Vector functions are used in a number of differential operations, such as gradient (measures the rate and direction of change in a scalar field), curl (measures the tendency of the vector function to rotate about a point in a vector field), and divergence (measures the magnitude of a source at a given point in a vector field).

• Vector Fields

  • Gradient field: Vector fields can be constructed out of scalar fields using the gradient operator (denoted by the del: ∇).
  • A vector field V defined on a set S is called a gradient field or a conservative field if there exists a real-valued function (a scalar field) f on S such that:

• Vector-Valued Functions

  • Also called vector functions, vector valued functions allow you to express the position of a point in multiple dimensions within a single function.
  • The input into a vector valued function can be a vector or a scalar.
  • A three-dimensional vector valued function requires three functions, one for each dimension.
  • This can be broken down into three separate functions called component functions:
  • This function is representing a position.

• Fundamental Theorem for Line Integrals

  • The gradient theorem, also known as the fundamental theorem of calculus for line integrals, says that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve.
  • The gradient theorem also has an interesting converse: any conservative vector field can be expressed as the gradient of a scalar field.
  • If $\varphi$ is a differentiable function from some open subset $U$ (of $R^n$) to $R$, and if $r$ is a differentiable function from some closed interval $[a,b]$ to $U$, then by the multivariate chain rule, the composite function $\circ r$ is differentiable on $(a,b)$ and $\frac{d}{dt}(\varphi \circ \mathbf{r})(t)=\nabla \varphi(\mathbf{r}(t)) \cdot \mathbf{r}'(t)$ for for all $t$ in $(a,b)$.
  • Electric field is a vector field which can be represented as a gradient of a scalar field, called electric potential.

• Stokes' Theorem

  • Therefore, electric field can be written as a gradient of a scalar field:
  • Since $\nabla \times \nabla f = 0$ for an arbitrary function $f$, we derive:
  • The scalar field $\varphi$ in the case of electromagnetism is called electric potential.