Vector calculus

[math]\displaystyle{ \newcommand{\uvec}{\overline{\textbf{u}}} \newcommand{\vvec}{\overline{\textbf{v}}} \newcommand{\evec}{\overline{\textbf{e}}} \newcommand{\Omegavec}{\overline{\mathbf{\Omega}}} \newcommand{\velang}[2]{\Omegavec^{\textrm{#1}}_{\textrm{#2}}} \newcommand{\ds}{\textrm{d}} \newcommand{\ts}{\textrm{t}} \newcommand{\us}{\textrm{u}} \newcommand{\vs}{\textrm{v}} \newcommand{\Rs}{\textrm{R}} \newcommand{\Ts}{\textrm{T}} \newcommand{\Ts}{\textrm{T}} \newcommand{\Bs}{\textrm{B}} \newcommand{\es}{\textrm{e}} \newcommand{\is}{\textrm{i}} \newcommand{\us}{\textrm{u}} \newcommand{\Os}{\textbf{O}} \newcommand{\Qs}{\textbf{Q}} \newcommand{\Ps}{\textbf{P}} \newcommand{\Ss}{\textbf{S}} \newcommand{\deg}{^\textsf{o}} \newcommand{\vec}[1]{\overline{#1}} }[/math]

V.1 Geometric representation of a vector

Vectors may be represented geometrically through an image, showing their direction with an arrow and their value with a scalar (positive or negative) (Figure V.1). Both direction and value may change with time.

Figure V.1 Geometric representation of the position vector of a point [math]\displaystyle{ \Os }[/math] moving along a straight line:
(a) geometric definition of a positive value; (b) three particular cases

The usual operations between vectors (addition, subtraction, scalar product, vector product, time derivative) may be carried out from their geometric representations. The next section summarizes the procedures.

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V.2 Operations between vectors with geometric representation

Instantaneous operations: addition, scalar product, vector product

Figure V.2 summarizes the procedures to carry out the three operations between vectors that involve just one time instant.

Figure V.2 Geometric calculation of instantaneous operations between vectors

Operations along time: time derivative

The two vector operations along time are the time derivative and the time integral, and they both depend on the reference frame where vectors are being observed. The time integral from the geometric representation is more complicated than the time derivative, and will not be presented. The time derivative of a vector relative to a reference frame R assesses the rate of change along time of the two vector characteristics (direction and value) between two close time instants, separated by a time differential (dt). The symbolic representation of that operation is [math]\displaystyle{ \frac{\ds\uvec}{\ds\ts}\bigg]_\Rs }[/math]. The subscript R reminds that the result depends on the reference frame from which the time evolution of the vector is observed.

The result of the time derivative is different from zero whenever the value, or the direction, or both characteristics change.

Many textbooks use a dot to indicate the time derivative of scalars and vectors:

Scalar variable: [math]\displaystyle{ \frac{\ds\rho}{\ds\ts}\equiv \dot{\rho} }[/math]

Vector variable: [math]\displaystyle{ \frac{\ds\uvec}{\ds\ts}\bigg]_\Rs\equiv \dot{\uvec}\bigr]_\Rs }[/math]

In this course, the dot is used mainly for the time derivative of scalars.

Particular case: Time derivative of a vector with constant direction

When only the value of a vector [math]\displaystyle{ \uvec }[/math] changes (that is, when its direction is constant with respect to the reference frame), its time derivative is a vector parallel to[math]\displaystyle{ \uvec }[/math] with a value equal to the change of its value u in a dt ([math]\displaystyle{ \ds\us/\ds\ts\equiv\dot\us }[/math]). As the size of an object at a given time is an invariant, that result does not depend on the reference frame (Figure V.3):

[math]\displaystyle{ \begin{equation} \frac{\ds\uvec}{\ds\ts} \bigg]_\Rs = \frac{\ds\uvec}{\ds\ts} = \dot{\us}\frac{\uvec}{|\uvec|}\end{equation} }[/math] , where [math]\displaystyle{ \frac{\uvec}{|\uvec|} }[/math] is the unit vector of the [math]\displaystyle{ \uvec }[/math] direction.

Figure V.3 The time derivative of a vector with constant direction and variable value is parallel to the vector, and it is independent from the reference frame: (a) sign([math]\displaystyle{ \dot\us }[/math]) = sign([math]\displaystyle{ \us }[/math]); (b) sign([math]\displaystyle{ \dot\us }[/math]) = -sign([math]\displaystyle{ \us }[/math])

Particular case: Time derivative of a vector with constant value evolving on a plane fixed to the reference frame

Let’s consider a vector [math]\displaystyle{ \uvec }[/math] evolving on a plane [math]\displaystyle{ \Pi }[/math] fixed to the reference frame R (vector with planar motion relative to R). If only its direction changes in R, its time derivative is a vector orthogonal to [math]\displaystyle{ \uvec }[/math] with value equal to the product of the vector value (u) and the rate of change (in a dt) of the orientation angle [math]\displaystyle{ \theta }[/math] of the vector on the [math]\displaystyle{ \Pi }[/math] plane, [math]\displaystyle{ \us\frac{\ds\theta}{\ds\ts}=\us\dot{\theta} }[/math] (Figure V.4).

Figure V.4 The time derivative of a vector with constant value and variable direction is orthogonal to the vector, and depends on the reference frame

The concept rate of change of the orientation ([math]\displaystyle{ \dot{\theta} }[/math]) calls for the previous introduction of the angle of orientation ([math]\displaystyle{ \theta }[/math]), defined on plane [math]\displaystyle{ \Pi }[/math] from a fixed direction in [math]\displaystyle{ \Pi }[/math] and vector [math]\displaystyle{ \uvec }[/math]. The orientation of that plane in R and the rate of change of orientation[math]\displaystyle{ \dot{\theta} }[/math] of [math]\displaystyle{ \uvec }[/math] may be combined in just one mathematical object: the angular velocity of the vector with respect to R. It is a vector orthogonal to the plane and with value [math]\displaystyle{ \dot{\theta} }[/math]. The direction of the vector is given by the screw rule (Figure V.5). The generic notation that will be used in this course for the angular velocity of an object relative to a frame R is [math]\displaystyle{ \Omegavec^{\textup{object}}_\Rs }[/math].

Figure V.5 Angular velocity of a vector with constant value relative to a reference frame

The time derivative can be written from that angular velocity vector as[math]\displaystyle{ \frac{\ds\uvec}{\ds\ts}\bigg]_\Rs = {\Omegavec}^{\uvec}_\Rs \times \uvec }[/math].

Particular case: Time derivative of a vector with constant vaue but with an orientation evolving in a general way relative to the reference frame

It can be proved that, when the vector [math]\displaystyle{ \uvec }[/math] to be differentiated does not evolve on a plane but has a 3D evolution, the result of the time derivative can be obtained in the same way through its angular velocity [math]\displaystyle{ \Omegavec^{\uvec}_\Rs }[/math] [Batlle, J.A., Barjau, A. (2020) chapter 1 in Rigid body kinematics. Cambridge University Press]. The determination of that angular velocity (units C1 i C2), however, is more complicated.

General case: Time derivative of a vector evolving in a general way relative to a reference frame R

If vector [math]\displaystyle{ \uvec }[/math] evolves in a general way relative to a reference frame R (that is, its value and direction change simultaneously), its time derivative is (Figure V.6):

[math]\displaystyle{ \frac{\ds\uvec}{\ds\ts}\bigg]_\Rs = [\text{change of value}]+[\text{change of direction}]_\Rs = \dot{\us}\frac{\uvec}{|\uvec|} + {\Omegavec}^{\uvec}_\Rs\times\uvec }[/math]

Figure V.6 Time derivative of a vector relative to a reference frame: general case

From that equation, it is easy to see that the difference between the time derivatives of the same vector relative to two different reference frames R1 and R2 is:

[math]\displaystyle{ \frac{\ds\uvec}{\ds\ts}\bigg]_\textrm{R1}-\frac{\ds\uvec}{\ds\ts}\bigg]_\textrm{R2} = (\Omegavec^\uvec_\textrm{R1}-\Omegavec^\uvec_\textrm{R2})\times \uvec }[/math]

It can be proved that [math]\displaystyle{ (\Omegavec^\uvec_\textrm{R1}-\Omegavec^\uvec_\textrm{R2}=\Omegavec^\textrm{R2}_\textrm{R1}) }[/math] (the proof is long and will be skipped). Thus, when the two frames do not have a relative rotation [math]\displaystyle{ (\Omegavec^\textrm{R2}_\textrm{R1}=0) }[/math], the time derivative of a vector is the same in both frames. Otherwise:

[math]\displaystyle{ \frac{\ds\uvec}{\ds\ts}\bigg]_\textrm{R1}=\frac{\ds\uvec}{\ds\ts}\bigg]_\textrm{R2} + \Omegavec^\textrm{R2}_\textrm{R1}\times \uvec }[/math]

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V.3 Analytical representation of a vector

A vector can also be represented analytically through its components in three independent space directions. The unit vectors of those directions are denoted as [math]\displaystyle{ (\evec_1, \evec_2, \evec_3) }[/math] and they define a vector basis. The vector can be expressed as a linear combination of those unit vectors, and the coefficients are the components of the vector in that vector basis:

[math]\displaystyle{ \uvec=\textrm{u}_1\evec_1+\textrm{u}_2\evec_2+\textrm{u}_3\evec_3 }[/math]

In this course, we will restrict ourselves to direct orthogonal bases (dextrorotatory orthogonal bases), that is, bases where the unit vector of direction 3 is the vector product of the unit vectors of directions 1 and 2: [math]\displaystyle{ \evec_1\times \evec_2=\evec_3 }[/math]. When it comes to representing a vector basis through an image, usually the three axes have an intersection point. That point is irrelevant, and it is not the “basis origin”: the concept “origin” does not apply to vector bases (what defines a vector basis are the three directions). A same vector may be projected on different bases, but that modifies neither its value nor its direction.

Figure V.7 shows the projection, on two different vector bases, of vector [math]\displaystyle{ \overline{\textrm{r}} }[/math], associated with the radius of a rotating platform with planar motion relative to R. The basis [math]\displaystyle{ (1,2,3) }[/math] does not change its orientation with respect to R, whereas the basis [math]\displaystyle{ (1',2',3') }[/math] changes its orientation relative to R but not relative to R’ (which is the platform).

Figure V.7 A same vector projected on two different vector bases

An alternative notation (that will be used in this course to express vectors projected on vector bases) consists on writing the components in a column, ordered according to the numbering of the directions of the basis:

[math]\displaystyle{ \left\{\overline{\textbf{r}}\right\}_{\textrm{123}}\equiv \left\{\overline{\textbf{r}}\right\}_{\Bs}= \begin{Bmatrix}\textrm{r}cos\theta \\\textrm{r}sin\theta \\\textup{0} \end{Bmatrix} }[/math]

[math]\displaystyle{ \left\{\overline{\textbf{r}}\right\}_{\textrm{1'2'3'}}\equiv \left\{\overline{\textbf{r}}\right\}_{\textrm{B'}}= \begin{Bmatrix}\textrm{r} \\\textup{0} \\\textup{0} \end{Bmatrix} }[/math]

If the unit vectors have a constant direction relative to the reference frame, we say it is a fixed basis. Otherwise (when their direction changes with time), we say it is a moving basis. As the unit vectors are always orthogonal, we may talk about the orientation of the vector basis B, and the rate of change of that orientation relative to a reference frame R (or angular velocity of the basis relative to R),[math]\displaystyle{ \Omegavec^\Bs_\Rs }[/math].

In Figure V.7, the vector basis B = (1,2,3) does not change its orientation relative to R (it is a fixed basis in R), whereas the vector basis B = (1',2',3') does change its orientation relative to R but not relative to R’ (it is a fixed basis in R’ but a moving one in R): [math]\displaystyle{ \velang{B}{R} = \vec 0 }[/math], [math]\displaystyle{ \velang{B'}{R} \not= \vec{0} }[/math], [math]\displaystyle{ \velang{B}{R'}\neq\vec{0} }[/math], [math]\displaystyle{ \velang{B'}{R'}=\vec{0} }[/math].

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V.4 Operations between vectors with analytical representation

Instantaneous operations: addition, scalar product, vector product

Instantaneous operations between vectors may be carried out through vector bases. As they are instantaneous, being fixed or moving bases is not relevant. What is absolutely necessary is that both vectors are projected on the same basis.

[math]\displaystyle{ \left\{\uvec\right\}_{\textrm{B}}= \begin{Bmatrix}\us_1 \\\us_2 \\\us_3 \end{Bmatrix} }[/math] , [math]\displaystyle{ \left\{\vvec\right\}_{\textrm{B}}= \begin{Bmatrix}\vs_1 \\\vs_2 \\\vs_3 \end{Bmatrix} }[/math]

[math]\displaystyle{ \left\{\uvec\right\}_{\textrm{B}}+ \left\{\vvec\right\}_{\textrm{B}}= \begin{Bmatrix}\us_1+\vs_1 \\\us_2+\vs_2 \\\us_3+\vs_3 \end{Bmatrix} }[/math]

[math]\displaystyle{ \left\{\uvec\right\}_{\textrm{B}}· \left\{\vvec\right\}_{\textrm{B}}= \us_1\vs_1+\us_2\vs_2+\us_3\vs_3 }[/math]

[math]\displaystyle{ \left\{\uvec\right\}_{\textrm{B}}\times \left\{\vvec\right\}_{\textrm{B}}= \begin{Bmatrix}\us_1 \\\us_2 \\\us_3 \end{Bmatrix} \times \begin{Bmatrix}\vs_1 \\\vs_2 \\\vs_3 \end{Bmatrix} = \begin{Bmatrix}\us_2\vs_3-\us_3\vs_2 \\\us_3\vs_1-\us_1\vs_3 \\\us_1\vs_2-\us_2\vs_1 \end{Bmatrix} }[/math]

Video V.1 Alroritme per al càlcul analític del producte vectorial

Operations along time: time derivative

Projecting a vector on a vector basis [math]\displaystyle{ (\evec_1, \evec_2, \evec_3) }[/math] means expressing it as an addition of three orthogonal vectors [math]\displaystyle{ \uvec=\sum_{\is}^{}\us_\is\evec_\is }[/math]. If it is a fixed basis in R (the reference frame where the operation is performed), those vectors have constant direction. Hence:

[math]\displaystyle{ \frac{\ds(\us_\is\evec_\is)}{\ds\ts} =\dot{\us}_\is\evec_\is }[/math] , [math]\displaystyle{ \left\{\left.\frac{\ds\uvec}{\ds\ts} \right]_{\Rs}\right\}_\Bs\equiv\frac{\ds}{\ds\ts}\left\{\uvec\right\}_\Bs= \begin{Bmatrix}\dot{\us}_1 \\\dot{\us}_2 \\\dot{\us}_3 \end{Bmatrix} }[/math]

If it is a moving basis in R, those vectors change its orientation according to an angular velocity [math]\displaystyle{ \Omegavec^\Bs_\Rs }[/math]:

[math]\displaystyle{ \frac{\ds(\us_\is \evec_\is)}{\ds\ts} =\dot{\us}_\is\evec_\is+\left.\us_\is\frac{\ds\evec_\is}{\ds\ts}\right]_\Rs=\dot{\us}_\is\evec_\is+\us_\is(\Omegavec^\Bs_\Rs\times \evec_\is) }[/math] ,

[math]\displaystyle{ \left\{\left.\frac{\ds\uvec}{\ds\ts} \right]_\Rs\right\}_\Bs = \frac{\ds}{\ds\ts}\left\{\uvec\right\}_\Bs +\left\{\Omegavec^\Bs_\Rs\right\}_\Bs \times \left\{\uvec\right\}_\Bs= \begin{Bmatrix}\dot{\us}_1 \\\dot{\us}_2 \\\dot{\us}_3 \end{Bmatrix} + \begin{Bmatrix}\Omega_1 \\\Omega_2 \\\Omega_3 \end{Bmatrix} \times \begin{Bmatrix}\us_1 \\\us_2 \\\us_3 \end{Bmatrix} }[/math]

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