Vector Calculus is the calculus of vector and scalar fields. It is therefore inherently in multiple dimensions. Just as ‘ordinary’
calculus can be used with algebraic
functions to determine such things as the
slope, maximum and minimum points, changes in concavity, average values, and sums, vector calculus can be used to determine such things with regard to multi-dimensional fields.
Obviously, a deep understanding of vector calculus requires a knowledge of the basics of vector algebra. It is essential to know that scalar fields have a magnitude at every point, while vector fields have a magnitude and direction at every point, and to know what dot products and cross products are.
The main operational difference between function calculus and vector calculus is that the standard derivative d/dx is replaced with the differential operator ∇, pronounced ‘del’ or in the olden days‘nabla.’
∇ = (i ∂/∂x + j ∂/∂y + k ∂/∂z) in Cartesian coordinates in three dimensions, and operates on whatever is to the right of it. In this notation, i, j, and k are the unit vectors in the x, y, and z directions respectively, and ∂/∂w is the partial derivative with respect to w.
If ∇ operates on a scalar function, then the result is a vector ∇s = v. This is called the ‘gradient of s,' also sometimes written ‘grad s.’ Physically, the gradient gives the direction of the maximum instantaneous increase of s at any point. If s were the three variable equation for the varying pressure in a material
(s = f(x,y,z)), for instance, then v would point in the direction of the most drastic infinitesimal increase in pressure at any point. v would then be a vector field. (v=f(x,y,z)i + g(x,y,x)j + h(x,y,z)k).
Thus, if you want to find out the change in the value of a scalar field when you move an infinitesimal amount in a certain direction, it is ∇s • dl, where dl = the infinitesimal displacement in the direction (i dx + j dy + k dz in cartesian coordinates). Then the total change in the value of the scalar field over a path L is
∫ ∇s • dl.
This is entirely analogous to ordinary function calculus, where the infinitesimal change in the value of a function is df/dx * dx, and the total change over an interval is ∫ df/dx * dx = ΔF. Likewise, a local maximum or minimum point of a scalar field is a point where ∇s = 0, again analogous to ordinary calculus.
∇ can operate on a vector function in two ways, consistent with the two ways that vectors can be multiplied. The dot product of ∇ with a vector is called the divergence. The result is a scalar ∇ • v = s, and is in a way a measure of how much the vector field seems to radiate from a single other point at the point in question.
The cross product of ∇ with a vector is called the curl. The result is a vector ∇ x v = t, and is in a way an indication of how the vector field seems to ‘flow’ around the point in question.
The divergence and curl are often seen under integrals, for it is often necessary to know the sum of the divergence over a volume, or the sum of a curl over a surface. There are two theorems that are of the utmost importance here:
The divergence theorem, also known as Green’s Theorem, states that
∫vol∇ • v = ∫surfv • da
where da = nunit da is the element of area in the direction of the outward pointing perpendicular unit vector, called the ‘normal vector.’
In words, the divergence theorem states that the integral over a volume of the divergence of a vector field is equal to the integral over the surface enclosing the volume of the dot product of vector field with the unit vector perpendicular to the surface at every point.
Stokes’ Theorem states that
∫surf∇ x v = ∫loopv • dl
where dl is the element of area along the direction of the loop.
In words, this means that the integral over a surface of the curl of a vector field is equal to the line integral over the boundary of the surface of the dot product vector field with the direction of the loop at every point.
There are higher derivatives in vector calculus as well. In particular, the divergence of a gradient is called the Laplacian and is written ∇2s. The Laplacian operates on a scalar and the result is a scalar. Conceptually, it can be thought of as a measure of the amount by which the direction of maximum instantaneous increase of a scalar field at a point radiates from the point. The gradient of a divergence is also occasionally used, ∇(∇ • v), but it does not have a specific name.
It is straight forward to extend vector calculus to apply to scalar and vector fields in more than three dimensions, for there is nothing mathematically unique about three orthogonal dimensions. It is also sometimes necessary to express the gradient, divergence, and curl in terms of non- Cartesian coordinate systems, such as spherical and cylindrical coordinates. This can be accomplished with the results of vector algebra, for the components of the coordinate system in question must be related to the Cartesian components, and is also straight forward. As an example, the divergence of a vector field in spherical coordinates v = (vr runit, vΘΘunit, vφφunit) is:
∇ • v = 1/r2 ∂/∂r (r vr ) + 1/rsinΘ ∂/∂Θ (sinΘ vΘ) + 1/rsinΘ ∂/∂φ vφ
Vector calculus is especially useful in physics, where scalar and vector fields abound.