Functions of Several Variables A function of several variables is - PDF document

Functions of Several Variables A function of several variables is just what it sounds like. It may be viewed in at least three different ways. We will use a function of two variables as an example. • z = f ( x, y ) may be viewed as a function of the two independent variables x , y . • It may be viewed as a function defined at different points ( x, y ) in the plane. • It may be viewed as a function whose domain is the set of vectors < x, y > or x i + y j . Limits of Functions of Several Variables We define a limit of a function of several variables essentially the same way we define a limit for an ordinary function: Definition 1 (Limit) . lim x → c f ( x ) = L if ∀ ǫ > 0 , ∃ δ > 0 such that | f ( x ) − L | < ǫ whenever 0 < | x − c | < δ . Definition 2 (Limit) . lim x → c f ( x ) = L if ∀ ǫ > 0 , ∃ δ > 0 such that | f ( x ) − L | < ǫ whenever 0 < | x − c | < δ . Properties of Limits Rule of Thumb: If a property of limits makes sense when translated to refer to a limit of a function of several variables, then it is valid for a function of several variables. For example, the limit of a sum will be the sum of the limits, the limit of a difference will be the difference of the limits, the limit of a product will be the product of the limits and the limit of a quotient will be the quotient of the limits, provided the latter limit exists. Continuity The definition of continuity for a function of several variables is es- sentially the same as the definition for an ordinary function. Definition 3 (Continuity) . A function f is continuous at c if lim x → c f ( x ) = f ( c ) . Definition 4 (Continuity for a Function of Several Variables) . A func- tion f is continuous at c if lim x → c f ( x ) = f ( c ) . As with ordinary functions, functions of several variables will generally be continuous except where there’s an obvious reason for them not to be. 1

2 Partial Derivatives For a function of several variables, we have partial derivatives with respect to each of its variables. The definition is based on the definition of an ordinary derivative. Definition 5 (Derivative) . Let f : R → R . d f f ( x + h ) − f ( x ) dx ( x ) = lim h → 0 . h ∂f Definition 6 (Partial Derivative) . Let f : R 2 → R . ∂x ( x, y ) = f ( x + h, y ) − f ( x, y ) f ( x, y + h ) − f ( x, y ) , ∂f lim h → 0 ∂y ( x, y ) = lim h → 0 . h h The obvious generalizations hold for functions with more than two independent variables. Calculation of Partial Derivatives Effectively, we calculate the partial derivative of a function with respect to one of its independent variables by acting as if the other independent variables were actually constants. Notation The following notations for the partial derivatives of a function z = f ( x, y ) are equivalent. f x = ∂f ∂x = ∂z ∂x = f 1 = D 1 f = D x f f y = ∂f ∂y = ∂z ∂y = f 2 = D 2 f = D y f Higher Order Derivatives Since a partial derivative is itself a function of several variables, it has its own partial derivatives. = ∂ 2 f ∂y∂x = ∂ 2 z ( f x ) y = f xy = f 12 = ∂ � ∂f � ∂y ∂x ∂y∂x � ∂f � = ∂ 2 f ∂x∂y = ∂ 2 z ( f y ) x = f yx = f 21 = ∂ ∂x ∂y ∂x∂y Changing the Order of Differentiation Theorem 1 (Clairaut’s Theorem) . If f xy and f yx are both continuous on a disk containing ( a, b ) , then f xy ( a, b ) = f yx ( a, b ) .

3 Proof. Let φ ( h ) = f ( x + h, y + h ) − f ( x, y + h ) − f ( x + h, y ) + f ( x, y ). The motivation comes from writing either f xy or f yx as a limit. We may write φ ( h ) = α ( y + h ) − α ( y ), where α ( t ) = f ( x + h, t ) − f ( x, t ). The Mean Value Theorem implies α ( y + h ) − α ( y ) = α ′ ( t ) h for some t between y and y + h . Since α ′ ( t ) = f 2 ( x + h, t ) − f 2 ( x, t ), we have φ ( h ) = [ f 2 ( x + h, t ) − f 2 ( x, t )] h . If we write β ( s ) = f 2 ( s, t ), then f 2 ( x + h, t ) − f 2 ( x, t ) = β ( x + h ) − β ( x ). Clairault’s Theorem β ( s ) = f 2 ( s, t ), f 2 ( x + h, t ) − f 2 ( x, t ) = β ( x + h ) − β ( x ). By the Mean Value Theorem, β ( x + h ) − β ( x ) = β ′ ( s ) h for some s between x and x + h . Since β ′ ( s ) = f 21 ( s, t ), we get f 2 ( x + h, t ) − f 2 ( x, t ) = f 21 ( s, t ) h , so φ ( h ) = f 21 ( s, t ) h 2 . Thus φ ( h ) = f 21 ( s, t ) → f 21 ( x, y ) as h → 0, since f 21 is continuous at h 2 ( x, y ). A similar calculation shows φ ( h ) = f 12 ( s, t ) → f 12 ( x, y ) as h → 0, h 2 showing f 12 ( x, y ) = f 21 ( x, y ). � Tangent Planes Consider a surface z = f ( x, y ) and suppose we are interested in the plane tangent to the surface at the point ( a, b, c ), where c = f ( a, b ). Since ∂z ∂x represents about how much z will change if x changes by 1 and y is fixed, here, and elsewhere as we look at tangent planes, tangent plane approximations and differentials, the partial derivative shown really means the partial derivative’s value at the relevant point, in this case ( a, b ) , it seems reasonable to expect the vector < 1 , 0 , ∂z ∂x > to be tangent to the surface. Similarly, it is reasonable to expect the vector < 0 , 1 , ∂z ∂y > to be tangent to the surface. Tangent Planes

4 � � i j k � � ∂z � � = − ∂z ∂x i − ∂z � 1 0 � We thus expect n = ∂y j + k to be a normal � � ∂x � � ∂z � � 0 1 � � ∂y � � vector to the tangent plane. We thus take n = < − ∂z ∂x, − ∂z ∂y, 1 > . We thus get < − ∂z ∂x, − ∂z ∂y, 1 > · < x − a, y − b, z − c > = 0 as an equation for the tangent plane, or − ∂z ∂x ( x − a ) − ∂z ∂y ( y − b ) + ( z − c ) = 0, or z − c = ∂z ∂x ( x − a ) + ∂z ∂y ( y − b ). This should be reminiscent of the Point-Slope Formula for the equation of a line. Tangent Hyperplanes It generalizes to ∂y y − b = � n ( x i − a i ) i =1 ∂x i as an equation for the hyperplane tangent to the hypersurface y = f ( x 1 , x 2 , . . . , x n ) at the point ( a 1 , a 2 , . . . , a n , b ). Tangent Plane Approximations and Differentials If we take z − c = ∂z ∂x ( x − a ) + ∂z ∂y ( y − b ) and solve for z , we get z = c + ∂z ∂x ( x − a ) + ∂z ∂y ( y − b ) This should be reminiscent of the Tangent Line Approximation for or- dinary functions. We may use this formula to approximate f ( x, y ) at a point ( x, y ) close to a point ( a, b ). Definition 7 (Differentials) . dx = ∆ x = x − a dy = ∆ y = y − b dz = ∂z ∂x ( x − a ) + ∂z ∂y ( y − b ) Differentials

5 We may use the differential dz to approximate the change ∆ z = ∆ f of a function f ( x, y ) if the independent variables x and y change by amounts dx and dy . This generalizes in the obvious way to functions of more than two variables. Differentiability Recall that for an ordinary function y = f ( x ) which was differen- tiable at a point, we found dy − ∆ y → 0 as ∆ x → 0. ∆ x We take the analogue of this as a definition of differentiability for func- tions of several variables. We state the definition for the case of a function of two variables; the variation for more variables should be obvious. Definition 8 (Differentiable) . We say a function f ( x, y ) is differen- dz − ∆ z (∆ x ) 2 + (∆ y ) 2 → 0 . � tiable at a point if (∆ x ) 2 + (∆ y ) 2 → 0 as � Differentiability (∆ x ) 2 + (∆ y ) 2 is the distance between ( x, y ) and the point � Recall in question. Effectively, we are defining a function of several variables to be differ- entialbe when an approximation using differentials is reasonable. We still need a reasonable way of determining whether a function is differentiable. This is given by the following theorem. Differentiability Theorem 2. If both partial derivatives of a function z = f ( x, y ) are continuous in some open disc { ( x, y ) : ( x − a ) 2 + ( y − b ) 2 < r } centered at ( a, b ) , then f ( x, y ) is differentiable at ( a, b ) . dz − ∆ z (∆ x ) 2 + (∆ y ) 2 → Proof. We need to show � (∆ x ) 2 + (∆ y ) 2 → 0 as � 0. � ∂z � ∂x ( x − a ) + ∂z We may write ∆ z − dz = f ( x, y ) − f ( a, b ) − ∂y ( y − b ) = f ( x, y ) − f ( a, y ) − ∂z ∂x ( x − a ) + f ( a, y ) − f ( a, b ) − ∂z ∂y ( y − b ). Proof