Vector Analysis
Vector analysis is the multi-dimensional analogue of single-variable calculus. It is used to represent (and analyze) the motions of the solar system, fluid flows, the flow of electric charge, and other phenomena.
In single-variable calculus, the derivative of a function f from the real numbers to the real numbers is its rate of change. For example, if f(t) represents the distance traveled at time t, the derivative of f at t represents speed. Vector analysis on the other hand typically concerns functions from three-dimensional space to the real numbers. Derivatives of such functions are taken with respect to a direction. For example, suppose that the function f represents the temperature of a room. Then its derivative in the upwards direction is the rate of temperature change as we move upwards. Often this direction is given a name like positive z-direction. Then the derivative of f in the positive z-direction is written as ∂f/∂z. In words, it is called the partial derivative of f in the z-direction. A point in three-dimensional space is typically represented by an ordered triplets of numbers called the coordinates of the point. The first number in the triplet is called the x-coordinate, the second is the y-coordinate, and the third is the z-coordinate. ∂f/∂z at a point (x1,y1,z1) then measures the rate at which the value f(x1,y1,z1) changes when the third coordinate z1 is changed. In other words, consider the single variable function g(z) = f(x1,y1,z1). Then ∂f/∂z = the derivative of g at z1.
The gradient of f at a point p is the vector given by the coordinates (∂f/ ∂x at p, ∂f/ ∂y at p, ∂f/ ∂z at p). Thus the gradient of f is a function from three-dimensional space to three-dimensional space. It is also called a vector field.
The pictures are to be interpreted thusly. At each point the value of the vector field is the vector whose tail is at that point, whose direction is given by the arrow and whose magnitude is the length of the arrow.
Suppose that a vector field represents the movement of a gas. That is, every vector shows the direction and speed of a particle of gas located near the vector's tail. The divergence of the vector field is a function that gives the "time rate of change of volume per unit volume" of the gas. Consider a small volume of gas near a point P in space. As time progresses, this gas will move in the direction that the vector field determines. If more gas is going out of this region around P than is coming in, then the divergence is negative. If less gas comes in then goes out, the divergence is positive. So the divergence measures the change in volume (per unit volume) as time progresses. A useful fact in hydrodynamics is that water is virtually incompressible and this means that if a vector field represents the flow of water then its divergence is practically zero at every point. Also, the divergence of electric field intensity is proportional to the charge density. If the vector field is represented by a function F such that F(x,y,z) = (I(x,y,z), J(x,y,z), K(x,y,z)) where I, J, and K are all functions of a single variable, then the divergence of F is equal to ∂I /partialx + ∂J/∂y + ∂K/∂z. The two top vector fields shown above have positive divergence everywhere and the two bottom vector fields have zero divergence everywhere (in other words they are solenoidal).
The curl of a vector field is another vector field. It is defined as follows. Imagine a small paddle (something like 4 rectangular blocks of wood glued along their edges so that from one perspective it looks like a plus sign with a small square hole in the middle). Now put this small paddle in the vector field near a point P. We imagine that the vector field represents the flow of a fluid. Then, depending on how the axis of the paddle is situated, it will spin. In general, there will be exactly one way to position the paddle to maximize the amount of spinning. In this position, the axis of the paddle is parallel to the curl vector at P. The direction of the curl vector is given by the right-hand rule. Position your right hand so that your thumb I parallel to the axis of the paddle and the fingers point in the direction of spinning. Then, your thumb points in the direction of the curl vector. The magnitude of the curl vector is twice the angular speed of the paddle. Precisely, the curl of at a vector field F is equal to (∂K/∂y - ∂J/∂z, ∂I/∂z - ∂K/∂x, ∂J/∂x - ∂I/∂y). Imagine that the vector fields in the picture above are extended to three-dimensional space in a such a way that their z-components are zero everywhere. Then the ones on the left have zero curl everywhere (in other words, they are irrotational). The curl of the vector field on the bottom right is pointing directly at the reader. The curl of the vector field on the top right is more complicated: it is zero on the two diagonal x = y and x = -y. It points directly at the reader when y is greater than x and directly away from the reader when x is greater than y.
In single-variable calculus, the integral of a function over the interval [a,b] on the real number line is its average value on that interval multiplied by the length (b-a) of the interval. The fundamental theorem of calculus states that the integral of the derivative of a function f over an interval [a,b] is equal to f(b) - f(a). For example, the distance traveled over an hour is equal to the average speed during that hour times one hour.
In vector analysis, the functions of interest might be defined on a surface or curve inside three-dimensional space. The integral of such a function over a finite curve or surface is its average value on the curve or surface multiplied by the curve's length or the surface's area. The fundamental theorem of vector analysis is called Stokes' theorem. It states that the integral of the (normal component of the) curl of a vector field over a bounded surface is equal to the integral of the (tangential component of the) vector field along the boundary of the surface.
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