Classical Mechanics

The Newtonian Approach

Dr. Cristian Giovanny Bernal, IMEF FURG

1.  Vector Calculus

1.1  Introduction and Basic Definitions

1.1.1  Scalars and Vectors

Physical quantities that are completely determined by the specification of one numerical value and a unit are called:
    Scalars     (e.g., mass, temperature, energy, wavelength).

Quantities that for a complete description besides the numerical value and the physical unit still need the specification of their direction are called:
    Vectors     (e.g., force, velocity, acceleration, torque).

A vector may be represented geometrically by an oriented distance, i.e., by a distance associated with a direction, such that holds; for example: Let A be the initial point and B the endpoint of the vector a.


Figure 1.1: Vector a pointing from A to B.

The magnitude of the vector is then represented by the length of the distance AB. A vector is frequently described symbolically by a Latin letter with a small arrow attached to elucidate the vector character. Other possible representations make use of German letters or emphasize the quantity by bold printing.
The magnitude of a vector a is written as:


1.1.2  Vector Algebra

Definition: Two vectors  a  and  b  are called equal (a = b) if:


That means: All distances of equal length and equal orientation are representations of the same vector on equal footing. Hence, the specific location of the vector in space is being disregarded.


Figure 1.2: The vectors a and b are equal.

A vector with opposite direction but equal magnitude of a is denoted as −a. Oppositely equal vectors have the same length and are located on parallel straight lines but have opposite orientations; that is, they are antiparallel:


If, for instance,


Addition: If two vectors a and b are added, the initial point of the one vector is brought by a parallel shift to coincide with the endpoint of the other one. The sum a + b, also called the resultant, then corresponds to the distance from the initial point of the first vector to the endpoint of the second one. This sum may also be found as the diagonal of the parallelogram formed by a and b.


Figure 1.3: Addition of the vectors a and b.

Rules of calculation: There hold



Figure 1.4: Illustration of the commutativity of the addition of vectors (left), and the associativity of the addition of vectors (right).

Computational Demonstration 1.1: The commutativity of vector addition

Subtraction: The difference of two vectors a and b is defined as


Zero (Null) vector: The vector difference aa is denoted as zero vector (or null vector):


The zero vector has magnitude 0; it is orientationless.


Figure 1.5: The zero vector.

Multiplication of a vector by a scalar: The product pa of a vector a by a scalar p, where p is a real number, is understood as the vector having (or not) the same orientation as a and the magnitude


Rules of calculation: There hold (where p and q are real)


These rules are immediately intelligible and don’t need any further explanation.


Figure 1.6: The multiplication of a vector a by a scalar p (in this case, p = 3).

Example 1.1: A vacation trip

A car travels 20.0 km due north (vector A) and then 35.0 km in a direction 60.0° west of north (vector B). Find the magnitude and direction of the car’s resultant displacement (vector R).

Exercise 1.1: Vectorial Hubble’s law

Hubble found that distant galaxies are receding with a velocity proportional to their distance from us, where we are on Earth. He found that the for i-th galaxy (with us at the origin), MC-Chap1_17.png, where MC-Chap1_18.png is the Hubble constant. Show that this recession of the galaxies from us does not imply that we are at the center of the universe. Specifically, take the galaxy at MC-Chap1_19.png as a new origin and show that Hubble’s law is still obeyed.

Exercise 1.2: Find vectors

Show how to find a and b, given a + b  and a - b.

1.1.3  Cartesian Components

The vector a, which is uniquely represented by the sum of vectors, is called the linear combination of the vectors. According to this, the vector a then must be a linear combination of the vectors b, c, d; thus


MC-Chap1_21.png, MC-Chap1_22.png, and MC-Chap1_23.png are denoted as components of the vector a with respect to b, c, d. The vectors b, c, d must be linearly independent, that is, none of the three vectors may be represented by the other two vectors. Otherwise not every arbitrary vector a could be combined out of the three basic vectors b, c, d.

Computational Demonstration 1.2: 3D linear combination of vectors

Component representation of a vector in Cartesian coordinates:  Any vector of the three-dimensional space may be represented as a linear combination of the Cartesian unit vectors i, j, k. This representation leads to simple and transparent calculations, due to the orthogonality relations. One then has


where MC-Chap1_26.png, MC-Chap1_27.png, e MC-Chap1_28.png, are the projections of a onto the axes of the frame. The unit vectors i, j, k  (or MC-Chap1_29.png) are also called base vectors. Besides the representation as a sum of vectors along the unit vectors, the vector a still may be represented as


If the base vectors are known, it is sufficient to know the three components.

Calculation of the magnitude of a vector from the components: According to the theorem of Pythagoras, the magnitude of a vector  a is calculated from its Cartesian components as follows:


Calculation of the orientation of a 2D-vector from the components: According to the trigonometric tangent definition, we have:


Calculation of the components of a 2D-vector from the magnitude-angle: Using the trigonometric projections onto the coordinate axis, with φ  being the angle from +Ox, we obtain:


Addition of vectors expressed by components: One has


Here both commutativity as well as associativity of vector addition have been used repeatedly. Thus, the components of the sum vector are the sums of the corresponding components of the individual vectors.

The position vector: A point P in space may be uniquely fixed by specifying the vector beginning at the origin of the coordinate frame and pointing to the point P as endpoint.
The components of this vector, the position vector, then correspond to the coordinates (x, y, z) of the point P. Thus, for the position vector, which is mostly abbreviated by r, there holds



Figure 1.7: The position vector and its coordinates

Example 1.2: Superposition of forces

Four coplanar forces are acting at the point 0, as shown in the sketch. Calculate the net force F acting at the point 0!


Example 1.3: Addition and subtraction of vectors

A DC-10 “flies” 45º north-west at 930 km/h relative to ground. A strong breeze blows from the west with 120 km/h relative to ground. What are the velocity and direction of flight of the plane, assuming that there is no wind deflection?


Exercise 1.3: Distance vector

Calculate the length of the vector a that represents the distance vector between the points MC-Chap1_39.png and MC-Chap1_40.png.

Exercise 1.4: The sailboats

The velocity of sailboat A relative to sailboat B, MC-Chap1_41.png, is defined by the equation  MC-Chap1_42.png, where MC-Chap1_43.png is the velocity of sailboat A and MC-Chap1_44.png is the velocity of sailboat B.
Determine the velocity of
A relative to B and its orientation if:  MC-Chap1_45.png = 30 km/h east  and  MC-Chap1_46.png  = 40 km/h north.

Exercise 1.5: 3D Pythagoras’s theorem

By applying Pythagoras’s theorem (the usual two-dimensional version) twice over, prove that the length of a three-dimensional vector  r  satisfies:

1.2  The Scalar Product

1.2.1  Definition

The physical quantities force and path are oriented quantities and are represented by the vectors F and s. The mechanical work W performed by a force F along a straight path s is


where φ is the angle enclosed by F and s. W by itself, although originating from two vectors, is a scalar quantity. With a view on physical applications of this kind, we therefore define the scalar product.

The scalar product  (a · b) of two vectors is understood as


where φ is the angle enclosed by a and b. The product a · b is a real number. Expressed by words, the scalar product is defined as the magnitude of a multiplied by the projection of b onto a, or vice versa.


Figure 1.8: Illustration of the scalar product.

The visual meaning of the scalar product:

Magnitude of the projection of b onto a multiplied by magnitude of a, or

Magnitude of the projection of a onto b multiplied by magnitude of b.

Computational Demonstration 1.3:  Projection of the one vector onto another

1.2.2  Properties and rules

Properties of the scalar product:

a · b takes its maximum value for φ  equal to zero (cos 0 = 1, a parallel to b)


For φ  equal to π  the scalar product takes its minimum value (cos π = -1, a antiparallel to b), namely


For φ = π/2,  a · b = 0  holds, even if a and b are nonzero (cos π/2 = 0, a perpendicular to b); thus


Rules of calculation: The following are true


The first and last rules are immediately intelligible; the second rule is illustrated in the figure below. If b, c, a are not coplanar, the rule of distributivity may easily be visualized by a triangle located in space. The vector a  may easily be visualized by a pencil or a pointing rod (compare the figures!).


Figure 1.9: Illustration of the distributivity law in a plane (left), and the distributivity law in space (right).

Unit vectors: Unit vectors are understood as vectors of magnitude 1. If a 0, then


A possibility frequently used in physics is to assign a direction to a scalarly formulated equation by the unit vector. For example, the gravitational force is acting along the connecting line between the two masses M and m. F is the force applied by the mass M to the mass m. Hence it is acting toward the mass M.



Figure 1.10: The unit vector pointing from the big mass to the small mass is MC-Chap1_60.png.

Cartesian unit vectors: The unit vectors pointing along the positive x-, y-, and z-axes of a Cartesian coordinate frame are defined as follows:


There exist two kinds of Cartesian coordinate frames, namely right-handed frames and left-handed frames. We shall always use only right-handed frames in these lectures!


Figure 1.11: Right-handed and left-handed systems.

Orthonormality relations: We now consider the properties of the Cartesian unit vectors with respect to formation of scalar products. Since the enclosed angle is each a right one, the following relations hold:


These relations are combined by defining


and is called the Kronecker symbol. For the three-dimensional space, μ and ν are running from 1 to 3.

Computational Demonstration 1.4: The dot product is not a vector

The scalar product in component representation: One has


Taking into account the orthonormality relations, we then get


Finally, setting for the indices MC-Chap1_68.png,  for MC-Chap1_69.png, and for MC-Chap1_70.png, then one can write


Hence, the scalar product of two vectors may be evaluated simply by multiplying the corresponding components of the vectors by each other and summing over the three products.

The angle between two vectors: From the knowledge of the two possibilities for representing the scalar product


one obtains the following relation for the angle enclosed by a and b:


Example 1.4: The cosine theorem

The cosine law of plane trigonometry is obtained by scalar multiplication. Use a triangle with sides a, b, and c to demonstrate this theorem!

Example 1.5: Projection of a vector onto another vector

Given the vectors a, b and c:
what is the absolute value of the projection of the sum (
a + b) onto the vector c?

Exercise 1.6: Dot product

By evaluating their dot product, find the values of the scalar p for which the two vectors (in unitary notation) are orthogonal: a = i + pj  and b = i - pj. (Remember that two vectors are orthogonal if and only if their dot product is zero.) Explain your answers with a sketch.

Exercise 1.7: Finding angles

The vectors a and b, are given by a = 2i + 3j  and b = -i + 2j. Find the angle between them!

1.3  The Vector Product (Axial Vector)

1.3.1  Basic Notions

One may define a further product between vectors. Here a new vector arises that is defined as follows.

Definition: The vector product of two vectors  a  and  b is the vector


where  n  is the unit vector being perpendicular to the plane fixed by a and b, and pointing out of the plane as a right-handed helix when rotating the first vector of the product into the second vector. Note that the rotation has to be performed along the shortest path.
    The magnitude of the vector product is equal to the area of the parallelogram spanned by a and b, as is seen from the figure.



Figure 1.12: Geometrical interpretation of the absolute value of the vector product as area.

Computational Demonstration 1.5: The area of the parallelogram.

1.3.2  Properties and Cartesian Representation

Properties of the vector product:

a × b takes its maximum value for φ = π/2 (sin π/2 = 1, a perpendicular to b)


a × b vanishes for  φ = 0  (sin 0 = 0, a parallel to b), namely


  The formula also includes the special case a = b, thus



⊙ represents a vector perpendicular to the drawing plane and pointing out of the plane (arrowhead).

represents a vector perpendicular to the drawing plane and pointing into the plane (arrowbase).

Computational Demonstration 1.6: Vector product is perpendicular

Rules of calculation: The vector product has the following properties:


Vector products of the Cartesian unit vectors: There holds


This product satisfies the cyclic permutability. For an anti-cyclic permutation one has to multiply by the factor −1, for example, j × i = k .

Vector product in components: We now denote the Cartesian unit vectors by  MC-Chap1_85.png instead of  i, j, k.   Let be two arbitrary vectors a and b.


When forming the vector product of the two vectors a × b, one obtains


In matrix form: The vector product a × b (above) may now be written as a three-row determinant:


Note: If the two vectors of the cross product are equal, then the two lower rows of the determinant are also equal, and the vector product vanishes.

Representation of the product vector: As we already stated in the definition of the vector product, the magnitude of the product vector may be visualized by a distance but better by the area of the parallelogram formed by the vectors. This vector is not determined by its length and orientation only (such vectors are called polar vectors) but is called an axial vector. Under a space reflection, which is also called a parity transformation, a polar vector changes its sign:  a → −a.  An axial vector, on the contrary, remains unchanged:  a × b = (−a) × (−b).


Figure 1.13: Representation of an axial vector resulting from a counter-clockwise electrical current (a) and a clockwise current (b). (c) shows an electrical dipole moment vector. In (d), it is shown that the mirror image of an axial vector that is parallel to the mirror plane points in the direction opposite to that of the original vector.

Example 1.6: Product of vectors

Use methods of the vector product to calculate:
(a)    The vector (1,
a, b)  is perpendicular to the two vectors (4, 3, 0) and (5, 1, 7). Find a and b.
(b)    Evaluate in Cartesian coordinates the vector product
a × b for a = (1, 7, 0) and b = (1, 1, 1).
(c)    Show that

Example 1.7: Force and torque

The following external forces are acting on a body:  

(a) Find the components, magnitude, and orientation of the resulting force
(b) Find the torque
M with respect to MC-Chap1_92.png.

Exercise 1.8: Cross product

Show that if  a = b + λc,  for some scalar λ, then a × c = b × c.

Exercise 1.9: Area of the parallelogram

Find the area A of the parallelogram with sides  a = i + 2j + 3k  and  b = 4i + 5j + 6k.

Exercise 1.10: kinetic energy

The kinetic energy of a single particle is given by MC-Chap1_93.png. For rotational motion this becomes MC-Chap1_94.png. Show that  MC-Chap1_95.png. For (r · ω) = 0 this reduces to MC-Chap1_96.png, with the moment of inertia I given by  MC-Chap1_97.png.

1.4  The Triple Scalar and Vector Product

1.4.1  Triple Scalar Product

Definition: The triple scalar product of the three vectors a, b, and c is defined as


that is, a combination of a scalar and vector product. The triple scalar product is therefore also denoted as a mixed product. The triple scalar product is a scalar.

Triple scalar product in component representation:


The three terms may again be combined to a determinant, such that


1.4.2  Properties

Cyclic permutability: The factors of the triple scalar product may be permuted cyclically. One has


These rules may be confirmed easily by successive permutations of the rows in the determinant above.

Geometry: the triple scalar product represents the volume of a parallelepipedon formed by the three vectors (see figure).



Figure 1.14: Illustration of the triple scalar product.

The volume has a positive sign (+) if a lies on the side of b × c, but a negative sign (−) if a lies on the side of b × c . Hence the volume might be associated with a sign. In general, however, this choice is not used, and a positive sign is always required. This is achieved by the definition  V = | a · (b × c) |.

Computational Demonstration 1.7: The parallelepipedon as a mixed product.

Properties of the triple scalar product:


that is, the three vectors are coplanar or (and) two vectors lie on a straight line. This is again a very clear proof of the theorems on determinants already mentioned above:

If two row vectors (or column vectors) are equal or proportional to each other, then the determinant equals zero.

When we permute two neighboring rows, the determinant changes by a factor (−1).

1.4.3  Vector Triple Product

By the vector triple product of three vectors a, b, c we mean the vector a × (b × c). Clearly, a × (b × c) is perpendicular to a and lies in the plane of b and c and so can be expressed in terms of them.
We note that the vector triple product is not associative, i.e.


Two useful formulae involving the vector triple product are


which may be derived by writing each vector in component form. It can also be shown that for any three vectors a, b, c .


This is Jacobi’s identity for vector products; for commutators it is important in the context of Lie algebras.

1.5  Differentiation and Integration of Vectors

1.5.1  Differentiation

Formation of the differential quotient: The vector A may occur as a function of a parameter. Let’s consider, for example, the position vector r(t) that—as a function of the time t—describes the path of a mass point. If one decomposes A into its components with respect to fixed unit vectors, then these components are functions of the parameter. We write


The differential quotient of a vector is formed by differentiating its components separately, as corresponds to the differentiation rule for sums. Because the unit vectors are not variables, they are conserved under differentiation,


By comparing the above relations, one notices that the differentiation of a vector in an arbitrary coordinate frame with fixed unit vectors amounts to the differentiation of the components of the vector. Generally, the rule for the n-fold differentiation of a vector reads


Differentiation of the product of a scalar and a vector:


This yields


Differentiation of the scalar product:


and therefore


Differentiation of the vector product:

It is performed analogously to the differentiation of the scalar product. Because the vector product is not commutative, one has to take care of the ordering of the factors.


This is easily proved by checking the individual components (e.g., the x-component) on both sides of the equation.

Example 1.8: Differentiation of the product of a scalar and a vector

For the scalar function φ(x) = x + 5 and the vector function A(x) = (MC-Chap1_117.png + 2x + 1, 2x, x + 2) the second derivative of the products φ · A is to be calculated.

Exercise 1.11: Differentiation of a vector

Find the second derivative of vector function  MC-Chap1_118.png

Application: Position, velocity, and acceleration of a mass point on a defined trajectory may be represented as vectors. The position vector for the motion of the mass point on an arbitrary trajectory B is the vector from the origin of the coordinate frame to the mass point; the variation of the position of the mass point with the time may be represented as time variation of the position vector (compare with the figure).



Figure 1.15: Definition of the orbital velocity: v = dr/dt.

The velocity vector is defined as the first derivative of the position vector r(t) of the orbital curve with respect to the time. The acceleration vector is obtained as the first derivative of the velocity with respect to the time, or as the second derivative of the position vector with respect to the time:


Note: Because the position vector is a vector, its derivatives with respect to the scalar time (t) are again vectors. Thus, the velocity and acceleration are vectors, too.

Computational Demonstration 1.8: Kinematics of a point in curved trajectory

Example 1.9: Velocity and acceleration on a space curve

Let the position vector be given by
Find the velocity and the acceleration as well as their magnitudes for the time points
t = 0 s  and  t = 1 s.


Example 1.10: The motion on a helix

The Cartesian coordinates of the helix read:
Find the vector position and calculate the velocity and acceleration vectors, as well as its magnitudes.


Computational Demonstration 1.9: The helix motion in 3D

Exercise 1.12: Circular motion

The Cartesian components of a circular motion are given by
ω  is the so-called angular velocity or also angular frequency. It is related to the revolution period
ω T = 2π.
(a) Write the position vector as a linear combination of unit vectors.
(b) Find the velocity vector and its magnitude. Prove that this vector is perpendicular to the position vector.
(c) Find the acceleration vector ans its magnitude. Identify this vector with the concept of centripetal acceleration.

1.5.2  Integration

The integration rules may be applied also to vectors in the customary way. For a vector A depending on a parameter (e.g., u), it follows that


Example 1.11: Integration of a vector I

Integrate the vector function MC-Chap1_130.png.

Exercise 1.13: Integration of a vector II

Integrate the vector function MC-Chap1_131.png, in the range [0, 2].

1.6  Coordinate Frames

1.6.1  Essentials

In an n-dimensional space one may always define n linearly independent base vectors out of which any arbitrary vector may be composed by a linear combination. For the sake of simplicity, vectors of magnitude unity are usually adopted as base vectors. A vector in the n-dimensional space reads


where the n base vectors MC-Chap1_133.png again shall satisfy the orthonormality relation. The scalar product of two n-dimensional vectors a and b may be defined by:


    The introduction of a coordinate frame implies that the coordinates of a space-fixed point change if the frame is displaced or rotated. From there it follows that for any special system a reference point  and a definite orientation  in space must be given. Physically seen, both quantities may be fixed by tying the coordinate frame.


Figure 1.16:  Special examples of the position of a point on an arbitrarily curved line (n  =  1),  The surface of a ball (n  =  2), and the space (n  =  3).

1.6.2  Transformation Equations

In order to change from one coordinate frame MC-Chap1_136.png to another one (here specifically the Cartesian frame: x, y, z), the following equations have to be set up


Cartesian coordinates: Given are the three base vectors MC-Chap1_138.png along the directions of three mutually perpendicular axes. The coordinates x, y, z of a point P  are the projections of the position vector MC-Chap1_139.png  onto the axes,


By convention the three unit vectors form a right-handed frame. Because they are mutually perpendicular, they constitute an orthogonal frame. Moreover, the unit vectors are always parallel to the axes, that is, fully independent of the position of the point P  in space.
     This constancy of direction of the unit vectors combined with their orthogonality is the reason for preferred usage of Cartesian coordinates. For many special problems with particular symmetry, it turns out as convenient to use coordinate frames that are adapted to the geometric conditions and therefore simplify the calculations.


Figure 1.17: The definition of Cartesian coordinates.

1.6.3  Curvilinear Coordinate Frames

In geometry, curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved. Commonly used curvilinear coordinate systems include: Cartesian (Rectangular), Spherical, and Cylindrical coordinate systems.
These coordinates may be derived from a set of Cartesian coordinates by using a transformation that is locally invertible (a one-to-one map) at each point. This means that one can convert a point given in a Cartesian coordinate system to its curvilinear coordinates and back.
The name curvilinear coordinates, coined by the French mathematician Lamé, derives from the fact that the coordinate surfaces of the curvilinear systems are curved.

Computational Demonstration 1.10: Coordinate systems

Note: Less commonly used, but of great importance for some fields of physics, other coordinate systems include: bipolar cylindrical, bispherical, parabolic, confocal paraboloidal, conical, elliptic cylindrical, hyperspherical, oblate spheroidal, parabolic cylindrical, prolate spheroidal, and toroidal coordinates.

1.6.4  Cylinder Coordinates

In this case, the coordinates used are in the definition are: ρ → separation of the point from the z-axis, φ → angle between the projection of the position vector onto the x, y plane and the x-axis, z → length of the projection of the position vector onto the z-axis (as in the Cartesian frame).

The coordinate areas extend to infinity (see figure, showing limited sections) and are



Figure 1.18:  The definition of cylindrical coordinates.

Transformation equations:   From the figure one may directly read off the relations


or in detail


To ensure that one point cannot be characterized by distinct combinations of coordinates, we agree on the following restrictions:


    The representation is not completely unique since the angle remains indefinite for points with ρ = 0. But inversely—and this is the more important requirement—to each triple  ρ, φ, z   only one space point is associated.

Unit vectors for cylindrical coordinates:  They can be obtained by partial differentiation of  r  with respect to ρ, φ, z  and subsequent normalization,


A second way is a geometrical approach. From the figure, eρ and eφ may be projected onto the x, y-plane without any changes. One has:


For solving kinematic problems it is important to know the derivative of the unit vectors  with respect to time. Let the functions ρ(t), φ(t), z(t)  be known. The generalization of the chain rule for a function of several variables then yields,


The derivative of a vector  e  of constant magnitude has no component along the direction of  e  and hence must be perpendicular to it:


We still note that from now on we shall frequently abbreviate the time derivative of a quantity by a dot above this quantity, as, for example,


Velocity and acceleration in cylindrical coordinates:  Let a point move along a path described by the position vector r(t). One has


In cylindrical coordinates let  ρ(t), φ(t), z(t)  be given. The position vector is


Note: These base vectors are now not fixed but are coordinate-dependent by themselves. One has to take care in component representation: For instance one cannot simply differentiate r = (ρ, 0, z) ! In order to avoid errors, one has to write out the vector, as, for example,


This yields the velocity :



This yields the acceleration :


Hence, in the cylindric frame both the velocity and acceleration are composed of three components: a radial component, an azimuthal component, and a component in the z direction.


Figure 1.19:  Coordinate surfaces and coordinate lines for cylindrical coordinates.

Example 1.12: Velocity and acceleration in a cardioid

A particle moves with constant velocity v along the heart curve or cardioid (Greek kardia = heart) given by,
Find the acceleration
a, its magnitude, and the angular velocity. (Note that r denotes here the coordinate ρ  of the cylindrical coordinate frame.)


Exercise 1.14:    Unitary parallelepipedon I

Check that the unit vectors of the cylindrical system build a unitary parallelepipedon.

Problem 1.15: Representation of a vector in cylindrical coordinates

Write the vector MC-Chap1_162.png in cylindrical coordinates.

Problem 1.16: Cylindrical vector position

Check that in 3D cylindrical coordinates the position vector r can be written with only two components, as above.

1.6.5  Spherical Coordinates

According to the figure below, the coordinates are


The point P is the intersection point of a circular cone about the z-axis with the vortex at the origin 0, a half-plane including the z-axis, and a sphere with the center at 0 that results by keeping the radius r constant and varying the two angles.


Figure 1.20:  The definition of spherical coordinates.

Transformation equations:   From the figure one may directly read off the relations


When the equations are written in detail, we get


To reach uniqueness, the following restrictions are agreed upon:


Unit vectors  for spherical coordinates:  The position vector r is,


In this case, partial differentiation of the position vector yields,


The unit vectors follow by normalization:


Note: The unit vector er points along the position vector, that is, it is the normal to the surface of the sphere. The unit vector eϑ is parallel to the x, y-plane, and the unit vector eφ has a component sin ϑ  along the negative z-direction.

Velocity and acceleration in spherical coordinates:  To calculate the velocity and acceleration in spherical coordinates, we still need the time derivatives of the unit vectors. One finds


and similarly


  Now we may calculate the velocity and acceleration in spherical coordinates. The following hold:

If ϑ = π/ 2, that is,  sin ϑ = 1,  MC-Chap1_174.png,  cos ϑ = 0,


respectively. These expressions for velocity and acceleration in plane polar coordinates are already known from the discussion on cylinder coordinates.


Figure 1.21:  Coordinate surfaces and coordinate lines for spherical coordinates.

Example 1.13: Angular velocity and radial acceleration

A rod rotates about MC-Chap1_177.png in a plane with the angular velocity ω. At the time t = 0, let φ = 0. The straight line intersects a fixed circle of radius a at the point MC-Chap1_178.png.
(a) Find the angular acceleration of the rod.
(b) Find the velocity
MC-Chap1_180.png and the acceleration MC-Chap1_181.png of the point MC-Chap1_182.png along the rod.
(c) Find the velocity and the acceleration of the point
MC-Chap1_183.png with respect to the center of the circle.


Problem 1.17: Unitary parallelepipedon II

Check that the unit vectors of the spherical system build a unitary parallelepipedon.

Problem 1.18: Unit vector transformation

Use the Kramer’s rule to find the rectangular unit vectors MC-Chap1_185.png in function of the spherical unit vectors MC-Chap1_186.png.

1.7  Vector Differential Operations

1.9.1  Operators and Fields

A vector operator is a differential operator used in vector calculus with applications in physics. Vector operators are defined in terms of del, and include the gradient, divergence, and curl.

    Scalar fields:         The notion of scalar field means a function φ(x, y, z) that assigns a scalar, the value MC-Chap1_187.png, to any space point MC-Chap1_188.png. Examples are temperature fields T(x, y, z) and density fields ρ(x, y, z)  (e.g., mass density, charge density).

    Vector fields:        A vector field correspondingly means a function A(x, y, z) that assigns a vector MC-Chap1_189.png to any space point MC-Chap1_190.png. Vector fields are, for instance, electric and magnetic fields, characterized by the field strength vectors E  and B , or velocity fields v(x, y, z)  in flowing liquids or gases.

1.9.2  Gradient

Given a scalar field  φ(x, y, z), the gradient of the scalar field at a fixed position MC-Chap1_191.png, denoted by  grad MC-Chap1_192.png, is a vector pointing along the steepest ascent of φ, the magnitude of which equals the change of φ  per unit length of the path along the maximum ascent at the point  MC-Chap1_193.png.
     In this way, any point of a scalar field can be associated with a gradient vector. The set of gradient vectors forms a vector field associated to the scalar field. Mathematically the so-defined vector field is given by the relation


To simplify the mathematical description, the following notation is used:


Definition of an operator:  (∇ : spoken “nabla” or “nabla operator”.) It is  a symbolic vector (vector operator) that, when applied to a function φ, generates the gradient of φ. Taken as such, the operator is meaningless; it has to operate on something, for example a scalar function  φ(x, y, z).

The total differential of φ:  Using the infinitesimal position vector dr(dx, dy, dz), we write the total differential of a scalar function as follows:


Equipotential surfaces: are surfaces on which the function φ  takes a constant value,  φ(x, y, z) = constant.
     As has been shown above, there is the relation


Because dφ represents the sum of the increases of φ in each direction dr, dφ = 0 means to stay on an equipotential surface. For this case, it holds that


where MC-Chap1_199.png lies in the equipotential surface ES. The scalar product MC-Chap1_200.png vanishes only then if the cosine of the enclosed angle vanishes (compare the figure), provided that ∇φ ≠ 0. This implies that ∇φ and MC-Chap1_201.png are perpendicular to each other. Thus the gradient of φ is always perpendicular to the equipotential areas and  always points in the direction of the strongest increase of φ.


Figure 1.22: Equipotential lines and the direction of the gradient.

Computational Demonstration 1.11: The gradient operator in 2D and 3D.

Example 1.14: The associated gradient field

Find the associated scalar field φ,  from the associated gradient field: MC-Chap1_204.png

Problem 1.19: Cartesian-spherical gradient

Given the scalar field MC-Chap1_205.png, find the gradient of φ in terms of a radial unit vector MC-Chap1_206.png.

1.9.3  Divergence

Contrary to the gradient operation, the divergence is applied to vector fields. Given a vector field MC-Chap1_207.png, we further imagine a cuboid-shaped “control volume”  (rectangular box) with the edge lengths Δx, Δy, Δz. The “vector flow” across an area represents the entity of vectors penetrating it perpendicularly, that is, the normal components of the vectors integrated over the entire area.
     Thus the “flow” (total flow) through an infinitesimally small volume (Δxdx, Δydy, Δzdz)  reads


The expression in brackets is called divergence of the vector field A:


Thus, the divergence represents the vector flow through a volume ΔV  per unit volume. It may also be written in the form


This last relation may be interpreted as analytic definition. As has been shown, it is identical with the geometric definition, namely:



Figure 1.23:  Illustration of the divergence as flow of the vector field through a volume.

While the argument of the gradient operation is a scalar, the divergence represents the scalar product of the operator ∇ and the vector A. Two important properties are:

1. For a vanishing divergence, the total flow  through an infinitesimal volume equals zero, that is, the in-flow just balances the out-flow.

2. If at some point of the vector field div A > 0, one says that the vector field there has a source; for div A < 0, one speaks of a sink  of the vector field. This is immediately clear from the definition of the divergence as net flow = outflow−inflow  per unit volume.

Computational Demonstration 1.12: The divergence of a vector field.

Example 1.15: The associated gradient field

Calculate the divergence of the field of the position vectors: MC-Chap1_214.png.

Problem 1.20: Divergence of a central force field

Calculate the divergence of the central force field given by: r·f(r), with MC-Chap1_215.png and f(r) being an arbitrary function of r.

1.9.4  Curl

The operation curl A  assigns a vector field curl A  to a given vector field A. The vector field curl A  informs about possible “vortices” of the field A  (a vortex exists if there is a closed curve in the vector field fulfilling the condition that the contour integral be zero —see theorem of Stokes).  The mathematical formulation of curl A  is given by


    The second definition states that the rotation may also be determined by forming the contour integral. The integration is performed over the vector field along a curve.

The rotation is thus determined by two distinct definitions (One has to prove that both definitions are identical). The first of these reads in detail:



Figure 1.24:  Illustration of a vector field A with vorticity on surface element ΔF with normal vector n.

Computational Demonstration 1.13: The curl of a vector field.

Example 1.16: The associated gradient field

Calculate the curl of the central force field given by: r·f(r), with MC-Chap1_220.png and f(r) being an arbitrary function of r.

Problem 1.21: Rotation of a vector field

Calculate the rotation of the vector field: MC-Chap1_221.png.

1.9.5  Multiple Application of the Vector Operator Nabla

Given a scalar field f(r)  and a vector field g(r), then



Thereby it is of course required that f is twofold continuously differentiable. The physicist always presupposes functions that are sufficiently often continuously differentiable; this is also assumed below. Hence, a gradient field has no vortices!



Hence, a rotation field has neither sources nor sinks, as is graphically clear: The vector field A = ω × r with ω = constant is so to speak an optimum vortex field (the velocity field of a rigid body rotating with the angular velocity ω).


Figure 1.25:  The velocity field of a rotating rigid body



1.10  Problems

1. Prove theorems:
The Graßmann expansion theorem (a), The Jacobi’s identity (b), The Lagrange’s identity (c):

2. The Frenet Formulas:
Deduce the three formulas of Frenet for the so-called the “moving trihedral” or “accompanying dreibein.” Apply them for a the circle, i.e. for a given position vector r(t), calculate the vectors of the moving trihedral.

3. Nabla in cylindrical and spherical coordinates:
Given a scalar field φ and a vector field A, deduce the relations, in cylindrical and spherical coordinates, for: MC-Chap1_230.gif

4. Electric field strength, electric potential:
Let a positive electric charge of magnitude Q be localized at the origin of the coordinate frame. The field intensity E describing the electrostatic field is given by,
where r denotes the spatial distance from the coordinate origin, and MC-Chap1_232.png represents the corresponding unit vector in radial direction. Calculate the associated potential field (let U denote the potential field, then E = −∇U) and show that it satisfies the Laplace equation ΔU = 0, except for the origin.

5. The constellation Ursa Major:
All the stars of the Big Dipper (part of the constellation Ursa Major) may appear to be the same distance from the earth, but in fact they are very far from each other. The figure  shows the distances from the earth to each of these stars. The distances are given in light-years (ly), the distance that light travels in one year. One light-year equals MC-Chap1_233.png.
(a) Alkaid and Merak are 25.6º apart in the earth’s sky. In a diagram, show the relative positions of Alkaid, Merak, and our sun. Find the distance in light-years from Alkaid to Merak.
(b) To an inhabitant of a planet orbiting Merak, how many degrees apart in the sky would Alkaid and our sun be?

1.11  In-Class Test 1

Created with the Wolfram Language