3.5: Vector functions, fields and calculus
Vectors can be parametrized, such that their values depend on a scalar parameter like time \(t\) or position on a curve \(s\). Such a parametrized vector, like \(\overrightarrow{\boldsymbol{F}}(t)\) or \(\overrightarrow{\boldsymbol{F}}(s)\) is also called a vector function . If a coordinate system is chosen, the vector function can be described by three scalar functions:
\[\overrightarrow{\boldsymbol{F}}(t)=F_{x}(t) \hat{\boldsymbol{\imath}}+F_{y}(t) \hat{\boldsymbol{\jmath}}+F_{z}(t) \hat{\boldsymbol{k}}. \nonumber\]
3.5.1 Vector and scalar fields
Even though they do not play an important role in this textbook we briefly mention the concept of vector fields. A vector field is a vector function that is defined at any point in (a region of) space. This is done by using the coordinates as parameters of the vector function, like in \(\overrightarrow{\boldsymbol{F}}(x, y, z)\). Examples of vector fields are the gravitational force field \(\overrightarrow{\boldsymbol{F}}(x, y, z)\) acting on a planet and the velocity field \(\overrightarrow{\boldsymbol{v}}(x, y, z)\) describing the velocities of all point masses in a rigid body. Instead of providing a single vector at one position, a vector field describes a field of vectors that are present at every location in space. Vector fields can also be time-dependent like in \(\overrightarrow{\boldsymbol{F}}(x, y, z, t)\). One can also define scalar fields, like \(V(x, y, z)\) which define a scalar value at every position. We will discuss scalar fields in more detail when dealing with potential energy in Ch. 7.
3.5.2 Calculus on vector functions
Vector calculus deals with methods to integrate or differentiate vector functions and fields. Although a complete discussion of vector calculus is outside the scope of this textbook, we say a few words about it here. Differentiation and integration over a vector functions can be performed by separating the vector into its components and then proceeding with each component like with scalar functions, e.g. for a time integral
\[\int \overrightarrow{\boldsymbol{F}}(t) \mathrm{d} t=\int F_{x}(t) \hat{\mathbf{z}} \mathrm{d} t+\int F_{y}(t) \hat{\boldsymbol{\jmath}} \mathrm{d} t+ \int F_{z}(t) \hat{\boldsymbol{k}} \mathrm{d} t. \nonumber\]
Taking derivatives is performed similarly
\[\frac{\mathrm{d}}{\mathrm{d} t} \overrightarrow{\boldsymbol{F}}(t)=\frac{\mathrm{d}}{\mathrm{d} t} F_{x}(t) \hat{\boldsymbol{\imath}}+ \frac{\mathrm{d}}{\mathrm{d} t} F_{y}(t) \hat{\boldsymbol{\jmath}}+\frac{\mathrm{d}}{\mathrm{d} t} F_{z}(t) \hat{\boldsymbol{k}}. \nonumber\]
Often Cartesian coordinates are to be preferred, because these operations become quite difficult in e.g. cylindrical coordinates where the unit vectors depend on position. However, when discussing kinematics we will deal with time derivatives of vector functions in cylindrical coordinates.
3.5.3 Calculus on scalar and vector fields
The differential operator, signified by the nabla symbol \(\overrightarrow{\boldsymbol{\nabla}}\), is important for analysing scalar and vector fields. In Cartesian coordinates it is defined as
\[\overrightarrow{\boldsymbol{\nabla}}=\frac{\partial}{\partial x} \hat{\boldsymbol{i}}+\frac{\partial}{\partial y} \hat{\boldsymbol{\jmath}}+\frac{\partial}{\partial z} \hat{\boldsymbol{k}}. \nonumber\]
The main operations that can be performed using \(\overrightarrow{\boldsymbol{\nabla}}\) are:
- The gradient \(\overrightarrow{\boldsymbol{\nabla}}\), e.g. \[\vec{\nabla} V(x, y, z)=-\overrightarrow{\boldsymbol{F}}(x, y, z), \nonumber\] which converts a scalar field to a vector field.
- The divergence, a dot product with \(\vec{\nabla}\), converts a vector field to a scalar field: e.g. \[\overrightarrow{\boldsymbol{\nabla}} \cdot \overrightarrow{\boldsymbol{v}}(x, y, z)=f(x, y, z). \nonumber\]
- The curl, a cross product with \(\vec{\nabla}\) converts a vector field to another vector field: e.g. \[\overrightarrow{\boldsymbol{\nabla}} \times \overrightarrow{\boldsymbol{v}}(x, y, z)=\overrightarrow{\boldsymbol{G}}(x, y, z). \nonumber\]
Vector and scalar fields can be integrated over a line in space, over a surface in space and over a volume in space using single, double or triple integrals. The detailed discussion of the multivariable calculus is outside the scope of this textbook, where we will only discuss the gradient and volume integrals over scalar fields.