Note how y and z don’t change in this limit. This kind of derivative is called a partial deriva- tive since only one of the variables changes. Fur- thermore, the concept of limit used here is just the Partial derivatives scalar limit you used in calculus, not the vector Math 131 Multivariate Calculus limit we discussed at the last meeting. There are D Joyce, Spring 2014 various notations for this limit, the most common being a variation of Leibniz’ notation, specifically Last time. We looked at the definition of the ∂f . The symbol ∂, which is read “partial” is a vari- multivariate limit lim f(x) = L as ∂x x→a ant of the letter d, and it’s only used to emphasize ∀ > 0, ∃δ > 0, ∀x ∈ Rn, that there are other partial derivatives besides the 0 < kx − ak < δ =⇒ kf(x) − Lk < ; one with respect to x. Thus, some topological concepts; properties of limits; con- tinuous functions; polynomials of several variables; ∂f f(x + h, y, z) − f(x, y, z) (x, y, z) = lim , component functions. ∂x h→0 h ∂f f(x, y + h, z) − f(x, y, z) The derivatives you know. Recall from calcu- (x, y, z) = lim , lus that if f : R → R is a scalar-valued func- ∂y h→0 h tion of one variable, usually denoted y = f(x), ∂f f(x, y, z + z) − f(x, y, z) (x, y, z) = lim . then its derivative at a scalar x, variously denoted ∂z h→0 h df(x) dy f 0(x), y0, , or , is defined in terms of limits Sometimes other notations for partial derivatives dx dx as are encountered. For instance, Dxf and fx are df(x) f(x + h) − f(x) ∂f = lim . sometimes used for the partial derivative . Note dx h→0 h ∂x 0 All the usual rules of differentiation followed from that primes, as in f can’t be used because they this definition. don’t indicate which variable is changing. Example 1. Let Partial derivatives of scalar fields. We’ll be- gin our generalization of derivatives by consider- f(x, y, z) = xz sin(3y + 4z). ing a scalar-valued function of several variables, f : Rn → R, also called scalar fields on Rn. Al- ∂f When you find , just treat y and z as constants. though f(x) = f(x1, x2, . . . , xn) depends on n dif- ∂x ferent variables, we can concentrate on what hap- Since xz sin(3y + 4z) is just x times some constant, pens when we change just one of them by leaving therefore its derivative is just that constant, that all the other variables fixed. That leads to the con- is, ∂f cept of “partial derivative.” Let’s take n = 3 so = z sin(3y + 4z). we don’t have to deal with subscripts, and then ∂x ∂f f(x) = f(x, y, z). Let’s let x be the variable that Finding is a little more work for this example we allow to change, and fix the other two variables ∂y y and z. Then we’ve converted f to a function of since the y is more deeply embedded in the mathe- just one variable, and we already have a concept of matical expression. In particular, the chain rule is derivative, namely, needed. f(x + h, y, z) − f(x, y, z) ∂f ∂ lim . = xz sin(3y + 4z) = xz cos(3y + 4z) 3. h→0 h ∂y ∂y
1 ∂f Finally, finding is even more work since the z surface z = f(x, y) at the point (a, b, f(a, b)) has ∂z the equation occurs twice in the expression. The function f is a product of two functions, namely xz and sin(3y + ∂f ∂f z = f(a, b)+ (a, b) (x−a)+ (a, b) (y−b). 4z), so we start by using the product rule ∂x ∂y
∂f In a different notation for partial derivatives, this ∂z equation becomes ∂ ∂ = xz sin(3y + 4z) + xz sin(3y + 4z) z = f(a, b) + fx(a, b)(x − a) + fy(a, b)(y − b). ∂z ∂z = x sin(3y + 4z) + xz cos(3y + 4z)4
There’s really nothing to computing partial derivatives; they’re just ordinary derivatives where only one variable actually varies. You’ll see lots more examples in the text and homework exercises.
Geometric interpretation of partial deriva- tives. Let’s find the geometric interpretation of a function f(x, y) of two variables. Our question is: ∂f (a, b) is a derivative, so it’s the slope of some ∂x tangent line. What tangent line? For such a function, its graph z = f(x, y) is a sur- face in R3. The point above (a, b) on that surface is (a, b, c), where the height above the (x, y)-plane is the value c = f(a, b) of the function at (a, b). What- Figure 1: Tangent plane to z = 1 − x2 − y2 ever the tangent line is, it passes through this point. The surface z = f(x, y) doesn’t have just a tangent Example 2. Let f(x, y) = 1 − x2 − y2. The graph line at (a, b, c), it has a whole tangent plane. But if of this function is a paraboloid shown in figure 1. we hold y fixed at the value b, that tangent plane 1 1 1 1 11 Let (a, b) = ( 4 , 2 ). Then f( 4 , 2 ) = 16 . We’ll find intersects the plane y = b in a tangent line, and the 1 1 11 ∂f the plane tangent to this surface at ( 4 , 2 , 16 ). slope of that tangent line is (a, b). The partial derivatives of f are fx(x, y) = −2x ∂x 1 1 Likewise the tangent plane intersects the plane and fy(x, y) = −2y. Their values at (a, b) = ( 4 , 2 ) 1 1 1 1 1 x = a in another tangent line, and the slope of that are fx( 4 , 2 ) = − 2 and fy( 4 , 2 ) = −1. So the slope ∂f of the tangent plane in the x-direction is − 1 , while tangent line is (a, b). 2 ∂y the slope of the tangent plane in the y-direction is −1. Therefore, the equation of the tangent plane is The tangent plane. We now have enough infor- z = f(a, b) + fx(a, b)(x − a) + fy(a, b)(y − b) mation to find the tangent plane. It’s a plane that = 11 − 1 (x − 1 ) − (y − 1 ) passes through the point (a, b, c) = (a, b, f(a, b)), 16 2 4 2 and we know the lines of intersection of the two planes x = a and y = b. That’s enough infor- Math 131 Home Page at mation to conclude that the plane tangent to the http://math.clarku.edu/~djoyce/ma131/
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