Concepts fundamental to the attitudes and methods of physical science
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Fields
Newton’s law of gravitation and Coulomb’s electrostatic law both give the force between two particles as inversely proportional to the square of their separation and directed along the line joining them. The force acting on one particle is a vector. It can be represented by a line with arrowhead; the length of the line is made proportional to the strength of the force, and the direction of the arrow shows the direction of the force. If a number of particles are acting simultaneously on the one considered, the resultant force is found by vector addition; the vectors representing each separate force are joined head to tail, and the resultant is given by the line joining the first tail to the last head.
In what follows the electrostatic force will be taken as typical, and Coulomb’s law is expressed in the form F = q1q2r/4πε0r3. The boldface characters F and r are vectors, F being the force which a point charge q1 exerts on another point charge q2. The combination r/r3 is a vector in the direction of r, the line joining q1 to q2, with magnitude 1/r2 as required by the inverse square law. When r is rendered in lightface, it means simply the magnitude of the vector r, without direction. The combination 4πε0 is a constant whose value is irrelevant to the present discussion. The combination q1r/4πε0r3 is called the electric field strength due to q1 at a distance r from q1 and is designated by E; it is clearly a vector parallel to r. At every point in space E takes a different value, determined by r, and the complete specification of E(r)—that is, the magnitude and direction of E at every point r—defines the electric field. If there are a number of different fixed charges, each produces its own electric field of inverse square character, and the resultant E at any point is the vector sum of the separate contributions. Thus, the magnitude and direction of E may change in a complicated fashion from point to point. Any particle carrying charge q that is put in a place where the field is E experiences a force qE (provided the other charges are not displaced when it is inserted; if they are E(r) must be recalculated for the actual positions of the charges).
A vector field, varying from point to point, is not always easily represented by a diagram, and it is often helpful for this purpose, as well as in mathematical analysis, to introduce the potential ϕ, from which E may be deduced. To appreciate its significance, the concept of vector gradient must be explained.
Gradient
The contours on a standard map are lines along which the height of the ground above sea level is constant. They usually take a complicated form, but if one imagines contours drawn at very close intervals of height and a small portion of the map to be greatly enlarged, the contours of this local region will become very nearly straight, like the two drawn in for heights h and h + δh.
Walking along any of these contours, one remains on the level. The slope of the ground is steepest along PQ, and, if the distance from P to Q is δl, the gradient is δh/δl or dh/dl in the limit when δh and δl are allowed to go to zero. The vector gradient is a vector of this magnitude drawn parallel to PQ and is written as grad h, or ∇h. Walking along any other line PR at an angle θ to PQ, the slope is less in the ratio PQ/PR, or cos θ. The slope along PR is (grad h) cos θ and is the component of the vector grad h along a line at an angle θ to the vector itself. This is an example of the general rule for finding components of vectors. In particular, the components parallel to the x and y directions have magnitude ∂h/∂x and ∂h/∂y (the partial derivatives, represented by the symbol ∂, mean, for instance, that ∂h/∂x is the rate at which h changes with distance in the x direction, if one moves so as to keep y constant; and ∂h/∂y is the rate of change in the y direction, x being constant). This result is expressed by
the quantities in brackets being the components of the vector along the coordinate axes. Vector quantities that vary in three dimensions can similarly be represented by three Cartesian components, along x, y, and z axes; e.g., V = (Vx, Vy, Vz).
Line integral
Imagine a line, not necessarily straight, drawn between two points A and B and marked off in innumerable small elements like δl in , which is to be thought of as a vector. If a vector field takes a value V at this point, the quantity Vδl·cos θ is called the scalar product of the two vectors V and δl and is written as V·δl. The sum of all similar contributions from the different δl gives, in the limit when the elements are made infinitesimally small, the line integral 
V ·dl along the line chosen.
Reverting to the contour map, it will be seen that (grad h)·dl is just the vertical height of B above A and that the value of the line integral is the same for all choices of line joining the two points. When a scalar quantity ϕ, having magnitude but not direction, is uniquely defined at every point in space, as h is on a two-dimensional map, the vector grad ϕ is then said to be irrotational, and ϕ(r) is the potential function from which a vector field grad ϕ can be derived. Not all vector fields can be derived from a potential function, but the Coulomb and gravitational fields are of this form.