Chapter 19

August 21 2017

Chapter 19 Partial Derivatives 19.1 Functions of Several Variables - many of the functions that arise in mathematics and its applications involve two or more independent variables - we have already met functions of this kind in our study of solid analytic geometry - so the equation z = x2 - y2 is the equation of a surface - but it also defines z as a function of the two variables x and y - and the surface can be thought of as the graph of this function - we usually denote an arbitrary function of the two variables x and y - by writing z = f(x, y) - we can visualize such a function by sketching its graph in xyz-space - picking a P = (x, y) as a suitable point in the domain of the function - the surface is thought of as lying "over" the domain - by obvious extension - T = f(x, y, z, t) - could be the temperature of a point inside an iron sphere - in this chapter we shall see the main themes of single variable differential calculus - derivatives, rates of change, chain rule computations, maximum-minimum problems, and differential equations - can all be extended to functions of several variables - however, there are striking differences between single variable calculus and multi variable calculus - since most of these differences already show up in functions of only two independent variables - we usually emphasize this case - and refer more briefly to functions of three or more variables - in the next chapter we turn to the integral calculus of functions of several variables - just as in our previous work - the domain of a function z = f(x, y) - is the set of all points P = (x, y) in the xy-plane - for which there exists a corresponding z - the domain is the largest set of points for which the formula makes sense - any finite combination of elementary functions is continuous at each point in its domain - a level curve lies in the domain of the function - and on it z = f(x, y) has constant value - a contour map is a collection of level curves - and can give a good idea of the shape of the graph - drawing graphs of functions of three variables is impossible - but we can use level surfaces - f(x, y, z) = c - to help get a useful intuitive idea of the nature of the function - applied to temperature and potential - they are isothermal and equipotential surfaces 19.2 Partial Derivatives - suppose y = f(x) is a function of only one variable - we know the derivative is - dy/dx = lim dx>0 f(x + dx) - f(x) / dx - and can be interpreted as the rate of change of y with respect to x - in the case of a function z = f(x, y) of two variables - we need similar mathematical machinery for working with the rate at which z changes as both x and y vary - the key idea is to allow only one variable to change at a time - while holding the other fixed - specifically - we differentiate with respect to only one variable at a time - regarding all others as constants - this gives us one derivative corresponding to each of the independent variables - these individual derivatives are the constituents from which we build the more complicated machinery that will be needed later - hold y fixed and letting x vary - dz/dx = lim dx > 0 f(x + dx, y) - f(x, y)/ dx - the limit is called the partial derivative of z with respect to x - the actual calculation of partial derivatives is very easy - treat every independent variable except the one we are interested in as if it were a constant - and apply the familiar rules - the partial derivatives of a function of x and y - are themselves functions of x and y - in the one variable case - we know the derivative dy/dx can legitimately be thought of as a fraction - the quotient of the differentials dy and dx - it is not possible to treat partial derivatives as fractions - when we are working with a function z = f(x, y) of only two variables - the partial derivatives have a simple geometric interpretation - dz/dx is the slope of the curve in the y = y0 plane at x = x0 - the pure second order partial derivatives - fxx = d2f / dx2, fyy = d2f / dy2 - don't represent anything new - each is found by holding one variable constant and differentiating twice with respect to the other variable - and each gives the rate of change of the rate of change of f in the direction of one of the axis - on the other hand - the mixed second partial derivatives - fxy = d2f/dydx, fyx = d2f/dxdy - the mxed partial derivative fxy gives the rate of change in the y-direction of the rate of change of f in the x-direction - it is not at all clear how these two mixed partials are related to each other - if both fxy and fyx exist for all points near (x0, y0) and are continuous at there - then fxy = fyx - in general, with suitable continuity - it is immaterial in what order a sequence of partial differentiations is carried out 19.3 The Tangent Place to a Surface - the concept of a tangent plane to a surface - corresponds to the concept of a tangent line to a curve - geometrically - the tangent plane to a surface at a point - is the plane that "best approximates" the surface near the point - for z = f(x, y) - the two partial derivatives fx and fy tangent lines determine a plane - if the surface is sufficiently smooth near the point - then this plane will be tangent to the surface at the point - to be quite clear - T is called the tangent plane at P0 - if as P approaches P0 - the angle between P0P and T approaches zero - we assume the tangent plane exists at P0 - and we develop a method of finding its equation - since P0 = (x0, y0, z0) lies on this tangent plane - we know the equation has the form - a (x - x0) + b (y - y0) + c (z - z0) = 0 - where N = ai + bj + ck is any normal vector - it remains to find N - to find this we use the cross product of two vectors V1 and V2 - that are tangent to the curves of x = x0 and y = y0 at P0 - to find V1 - we use the fact that along the tangent line to C1 of x = x0 - an increase of 1 unit in x produces a change fx in z - while y does not chagne at all - so V1 = i + 0 dot j + fx k - V2 = 0 dot i + j + fy k - so N = V2 x V1 - so the tangent plane equation is - z - z0 = fx (x - x0) + fy (y - y0) - an alternative method is to assume the given equations defines z implicitly - z - z0 = dz/dx (x - x0) + dz/dy (y - y0) - this method is of particular value when the equation of the surface is difficult or impossible to solve for z 19.4 Increments and Differentials. The Fundamental Lemma - most of calculus can be understood by using geometric intuition mixed with a little common sense - without getting bogged down in the underlying theory of the subject - in a few places - however - this theory is inescapable - because without it there is no way to grasp what is going on in the main developments of the subject itself - this is true for infinite series and the theory of convergence - it is also true for the topics of the next two sections - directional derivatives and the chain rule - for a function y = f(x) of one variable that has a derivative at a point x0 - if dx is an increment that carries x0 to a nearby point of x0 + dx - we are interested in the corresponding increment in y - dy = f'(x0)dx + err dx - where err > 0 as dx > 0 - the Fundamental Lemma extends this to multiple variables - suppose a function z = f(x, y) and its partial derivatives fx and fy are defined and continuous around a point - then - dz = fx dx + fy dy + err 1 dx + err 2 dy - where err 1 and err 2 > 0 as dx and dy > 0 19.5 Directional Derivatives and the Gradient - let f(x, y, z) be a function defined throughout some region of three-dimensional space - let P be a point in this region - at what rate does f change as we move away from P in a specified direction - in the directions of the axes - we know the rates of change of f are given by the partial derivatives - but how do we calculate the rate of change of f if we move away from P in a direction that is not a coordinate direction - in analyzing this problem - we will encounter the very important concept of the gradient of a function - let P = (x, y, z) and - R = xi + yj + zk be the position vector of P - let the specified direction be given by a unit vector u - if we move from P to Q - so Q = (x + dx, y + dy, z + dz) - then f will change by df - if we now divide this change df - by the distance ds = |dR| between P and Q - then the quotient df/ds is the average rate of change of f - with respect to distance - as we move from P to Q - for example - if f is the temperature at P - then df/ds is the average rate of change of temperature along PQ - the limiting value of df/ds as Q approaches P - df/ds = lim ds > 0 df/ds - is the derivative of f at point P in the direction u - or the directional derivative of f - in the case of the temperature function - df/ds represents the instantaneous rate of change of temperature with respect to distance - roughly speaking, how fast is it getting hotter - at the point P as we move away from P in the direction specified by u - this is all very well - but how do we actually calculate df/ds in a specific case - to discover how to do this, assume the function has continuous partial derivatives - with this, the Fundamental Lemma enables us to write df as - df = df/dx dx + df/dy dy + df/dz dz + err 1 dx + err 2 dy + err 3 dz - where the errors > 0 as dx, dy, dz > 0, that is, as ds > 0 - dividing by ds now gives - df/ds = df/dx dx/ds + df/dy dy/ds + df/dz dz/ds - this formula should be recognized as a speical kind of chain rule - in the sense - that as we move along the line through P and parallel to u - f is a function of x, y, and z - where x, y, z are in turn functions of the arc length s - the formula shows how to differentiate f with respect to s - the first factor in each product - depends only on the function f and the coordinates of the point P at which the partial derivatives of f are evaluated - while the second factor is independent of f and depends only on the direction in which df/ds is being calculated - so the formula can be thought of and written as the dot product of two vectors - df/ds = (df/dx i + df/dy j + df/dz k) dot dR/ds - the first factor here is a vector called the gradient of f - denoted grad f - so df/ds = grad f dot u - and df/ds = |grad f| cos angle - since u can be chosen to suit our convenience - the single vector grad f contains within itself - the directional derivative of f at P in all possible directions - the directional derivative df/ds in any given direction is the scalar component of grad f in that direction - the vector grad f points in the direction in which f increases the most rapidly - the length of the vector grad f is the maximum rate of increase of f - the gradient of a function of f(x, y, z) at a point P0 is normal to the level surface f that passes through P0 - N = grad f - is normal to the tangent plane of the level surface - the equation of the tangent plane is - df/dx (x - x0) + df/dy (y - y0) + df/dz (z - z0) = 0 - the main uses of direction derivatives and gradients are found in the geometry and physics of three-dimensional space - however - thees concepts can also be defined in two dimensions - and they have similar but thinner properties - so a curve f(x, y) = c0 can be thought of as a level curve of z = f(x, y) - and if the gradient of this function is - grad f = df/x i + df/dy j - then the value of this gradient at a point on the curve is a vector that is normal to the curve - the gradient of a function f(x, y, z) can be written in "operational form" as - grad f = (d/dx i + d/dy j + d/dz k) f - this is usually denoted by del f - the del operator is similar to - but more complicated than - the familiar operator d/dx - when del is applied to a function f - it produces a fector - namely the vector grad f 19.6 The Chain Rule for Partial Derivatives - the single variable chain rule for ordinary derivatives tells us how to differentiate composite functions - we know from experience this is an indispensable tool - the simplest multi variable chain rule involves a function w = f(x, y) - where x and y are each functions of another variable t - x = g(t) and y = h(t) - then w is a function of t - w = f [ g(t), h(t) ] = F(t) - the chain rule for this situation is - dw/dt = dw/dx dx/dt + dw/dy dy/dt - it is convenient to call w the dependent variable - x and y the intermediate variables - and t the independent variable - the right side has two terms - one for each intermediate variable - the formula extends in an obvious way to any number of intermediate variables - further - the intermediate variables can be functions of two or more varialbes - then the formula uses partial derivatives - in section 19.4 we defined the differential dw of w = f(x, y, z) as - dw = dw/ds dx + dw/dy dy + dw/dz dz - the chain rule tells us if x, y, z are themselves functions of t - then it is permissible to calculate dw/dt by formally dividing by dt - the individual terms - dw/dx dx are called the partial differentials of w with respect to x, y, z - so dw is the total differential 19.7 Maximum and Minimum Problems - in the case of functions of a single variable - one of the main applications of derivatives is the study of maxima and minima - such problems for functions of two or more variables can be much more complicated - if P = (x0, y0) has a maximum value for z = f(x, y) in its domain - then dz/dx = 0 and dz/dy = 0 - along the planes x0 and y0 - there are two equations with two unknowns, x0 and y0 - in many cases we can solve these equations simultaneously - but it is important to remember - the solution could be a saddle point - where the function is a maximum in one direction - and a minimum in the other - we would need to analyze this critical point to determine what significance it has - the second derivative test can be used to assess the critical point - D = fxx fyy - fxy2 - then the point is - a maximum if D > 0 and fxx < 0 - a minimum if D > 0 and fxx > 0 - a saddle point if D < 0 - undetermined if D = 0