**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