SIMPLE MULTIVARIATE CALCULUS

1. R EAL - VALUED F UNCTIONS OF S EVERAL VARIABLES 1.1. Definition of a real-valued function of several variables. Suppose D is a set of n-tuples of real numbers (x1 , x2 , x3 , . . . , xn ). A real-valued function f on D is a rule that assigns a unique (single) real number y = f (x1, x2, x3 , . . . , xn) to each element in D. The set D is the function’s domain. The set of y-values taken on by f is the range of the function. The symbol y is the dependent variable of f, and f is said to be a function of the n independent variables x1 to xn . We also call the x’s the function’s input variables and we call y the function’s output variable. A real-valued function of two variables is just a function whose domain is R2 and whose range is a subset of R1, or the real numbers. If we view the domain D as column vectors in Rn, we sometimes write the function as   x1  x2    f .   ..  xn 1.2. Examples. a: The volume of a right circular cylinder is a function of the radius and height, V = f(r, h) or V = πr2 h. A cylinder is represented in figure 1. The volume of the cylinder as a function of its radius and height is shown graphically in figure 2. Notice that the volume increases as both the radius and the height increase. F IGURE 1. A Cylinder

Date: October 18, 2005. 1

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F IGURE 2. The Volume of a Cylinder as a Function of Radius and Height 3

h 2 1 0 80 60 VHr,hL 40 20 0

0 1 r

2 3

b: The level of production from a given technology as a function of two inputs x1 and x2 is represented by y = f(x1 , x2 ). For example, a quadratic function might be f(x1 , x2) = 20x1 + 16x2 − 2x21 − x22. 1/4 1/2 A Cobb-Douglas function might be f(x1 , x2) = 10x1 x2 . We can represent the Cobb-Douglas function with a graph in three dimensions as in figure 3.

1/4 1/2

F IGURE 3. Cobb-Douglas Production Function f(x1 , x2) = 10x1 x2

100 fHx1 ,x2 L 50

30 20

0

x2 10

10 x1

20 30

1.3. Interior and boundary points in the plane or R2. 1.3.1. Interior points of regions in the plane (R2). A point (x01 , x02 ) in a region B in the x1 -x2 plane is an interior point of B if it is the center of a disk that lies entirely in B. 1.3.2. Boundary points of regions in the plane (R2 ). A point (x0 , y0 ) is a boundary point of B if every disk centered at (x0, y0) contains points that lie outside of B and well as points that lie in B. (The boundary point itself needs to belong to B). The interior points of a region, as a set, make up the interior of the region. The region’s boundary points make up its boundary.

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3

1.3.3. Open and closed regions in the plane (R2). A region is open if it consists entirely of interior points. A region is closed if it contains all of its boundary points. 1.3.4. Bounded regions in the plane. A region in the plane is bounded if it lies inside a disk of fixed radius. A region is unbounded if it is not bounded. Consider the region in the plane bounded by line segments in figure 4. Call this region the set B. The point a is an interior point because all points in a small disk centered at a lie within B. The point c is a boundary point because no matter how small the radius of the disk centered at c, it contains points both within and outside of B. The boundary of B consists of all points on the line segments defining B, and the interior of B consists of all points bounded by the line segments but which do not lie on them. The set B is a closed set. F IGURE 4. Interior and Boundary Points of a Region in the Plane

x2

c a

B

0

x1

Consider the region in the plane bounded by the dotted line in figure 5. Call this region B. The set consists all points within the dotted circle but does not include points on the circle. The point a is an interior point because all points in a small disk centered at a lie within B. The point c is a boundary point because no matter how small the radius of the disk centered at c, it contains points both within and outside of B. In this case the boundary point c is not contained in the set B. The boundary of B consists of all points on the circle defining B, and the interior of B consists of all points bounded by circle which do not lie on it. The set B is an open set. Consider the region in the plane on and to the northeast of the curved line in figure 6. Call this region B. The point a is an interior point because all points in a small disk centered at a lie within B. The point c is

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F IGURE 5. Interior and Boundary Points of a Region in the Plane x2

c a

B

0

x1

a boundary point because no matter how small the radius of the disk centered at c, it contains points both within and outside of B. The interior of B consists of all points to the northeast of the curved line in figure 6. The set B is an unbounded set. The set B is also a closed set. F IGURE 6. Interior and Boundary Points of a Region in the Plane

x2

B c

a

0

x1

1.4. Interior and boundary points in space or R3 . 1.4.1. Interior points of regions in space (R3). A point (x01 , x02 , x03 ) in a region D in space is an interior point of D if it is the center of a ball that lies entirely in D.

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5

1.4.2. Boundary points of regions in space (R3). A point (x01 , x02 , x03 ) is a boundary point of D if every sphere centered at (x01 , x02 , x03 ) encloses points that lie outside of D and well as points that lie in D. The interior of D is the set of interior point of D. The boundary of D is the set of boundary points of D. 1.4.3. Open and closed regions in space(R3). A region D is open if it consists entirely of interior points. A region is closed if it contains its entire boundary. 1.5. Interior and boundary points in Rn . We can extend the idea of open and closed sets to Rn . Let a = (a1,,a2,. . . ,an ) be a point in Rn and let r be a given positive number. The set of all points x ∈ Rn such that ( [x − a] · [x − a] )

1/2

< r 1/2

is called an open n-ball of radius r with center at a. We call ( [x − a] · [x − a] ) the distance between the point x and the point a. We denote this set by B(a;r). The ball B(a;r) consists of all points whose distance from a is less than r. In R1, this is an open interval with center at a. In R2, this is a circular disk with radius r and center at a. In R3 , this is spherical solid with center at a and radius r. 1.5.1. Interior points of sets in Rn. Let S be a subset of Rn , and assume that the point a is an element of S. Then a is called an interior point of S if there is an open n-ball with center at a, all of whose points belong to S. 1.5.2. Open sets in Rn . A set S in Rn is called open if all its points are interior points. 1.5.3. Closed sets in Rn. A set S in Rn is called closed if its complement Rn \ S is open. Consider the region in space consisting of the doughnut or torus in figure 7. Call this region B in figure 8. The point a is an interior point because all points in a ball or sphere centered at a lie within B as shown in figure 9. The point c is not an interior point because a sphere centered at point c would contain points both inside and outside the doughnut. F IGURE 7. A Subset of R3

1.6. Graphs and level curves for functions with domain in R2 . 1.6.1. The graph of a function with domain in R2. The set of all points {(x1 , x2 ), f(x1 , x2 )} in space (R3), for (x1 , x2 ) in the domain of f, is called the graph of f. The graph of f is also called the surface z = f(x1 , x2 ). Consider the function z = f (x1, x2) = 100 − x21 − x22 This function has a graph in R3 because its domain is R2 . The graph of f(x1 , x2) = 100 − x21 − x22 is shown in figure 10.

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F IGURE 8. The Interior of a Subset of R3

B a c

F IGURE 9. An Interior Point of a Subset of R3 is Contained within a Sphere within the Subset

B a c

1.6.2. The level curve of a function with domain in R2. The set of points in the plane (R2 ) where a function f(x1 , x2 ) has a constant value f(x1, x2 ) = c is called a level curve of f. One can think of a level curve like contour lines on a map. Points on the same curve or line represent the same height of the function. Level curves are created by intersecting a plane at a given height (or value of the function f(x1,x2 )) with the graph of f(x1,x2 ), noting the values of (x1,x2 ) where these intersections occur, and then plotting them in the x1 -x2 plane. As an example, consider the function z = f (x1, x2) = 100 − x21 − x22 from figure 10. In figure 11, we show both the graph of the function and a plane at function value of 30. A series of level curves for f(x1 , x2) = 100 − x21 − x22 at various valued of the function are contained in figure 12.

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F IGURE 10. Graph of the function z = 100 − x21 − x22

100 50 0 f Hx1 ,x2 L -50 -100

10 5 -5 0

0 x1

x2

-5

5 10 -10

F IGURE 11. Graph of the function f(x1 , x2) = 100 − x21 − x22 along with intersecting plane

100 50 f Hx1 ,x2 L

0

10

-10 10

5 -5

0 0 x1

-5

5 10

-10

1.6.3. Second example of a function with domain in R2. Consider the function

x2

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F IGURE 12. Contour lines for the function f(x1 , x2) = 100 − x21 − x22

10

5

0

-5

-10 -10

0

-5

z = f (x1 , x2) =

5

10

x21 x22 (1 + x21 ) (1 + x22)

The graph of the function and a plane at z = 2 is depicted in figure 13. A level curve for the function when z = 2 is given in figure 14. A more general set of level curves is depicted in figure 15.

F IGURE 13. Graph of the function z =

x21 x22 (1 + x21 ) (1 + x22 )

0 0.5 x1 1.5 2 3

2

f Hx1 ,x2 L

1

0 0

0.5

1

1.5

2

x2

1.6.4. Third example of a function with domain in R2. Consider the function −x1 x2 2 2 ex1 + x2 The graph of the function is depicted in figure 16. A set of level curves is depicted in figure 17. z = f (x1, x2) =

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F IGURE 14. Contour line for the function z =

9

x21 x22 (1 + x21 ) (1 + x22 )

when z = 2

x2 2

1.5

1

0.5

0 0

0.5

1

1.5

x1

2

F IGURE 15. Contour lines for the function z =

x21 x22 (1 + x21 ) (1 + x22 )

x2 2

1.5

1

0.5

0 0

0.5

1

1.5

2

x1

1.6.5. Functions with domain in R3 . The set of points (x1 , x2 , x3 ) in space where a function of three independent variables f(x1, x2 , x3 ) has a constant value f(x1 , x2 , x3 ) = c is called a level surface of f. The set of all points {x1 , x2 , x3 , f(x1 , x2 , x3 )} in space, for (x1, x2 , x3 ) in the domain of f, is called the graph of f. The graph of f is also called the surface w = f(x1, x2 , x3 ). One example of the level set for a function with a domain in R3 in contained in figure 18 while a second example is shown in figure 19. 2. L IMITS 2.1. Definition of a limit for a real valued function with a domain in R2 . Definition 1 (Limit). We say that a function f(x1 ,x2 ) approaches the limit L as (x1 , x2 ) approaches (x01 , x02 ), and write

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F IGURE 16. Graph of the function f(x1 , x2) =

−x1 x2 2 2 ex1 + x2

0.2 0.1 0 fHx1,x2L -0.1 -

-2 2 -1

-0.2 2-

x1

1

0 0

1 -1

x2

2 -2

F IGURE 17. Levels curves for the function f(x1 , x2) =

−x1 x2 2 2 ex1 + x2

x2 2

1

0

-1

-2 -2

-1

lim

0

(x1 , x2 ) → (x01 , x02 )

1

2

x1

f(x1 , x2) = L

if, for every number ε > 0, there exists a corresponding number δ > 0 such that for all (x1 ,x2 ) in the domain of f, 0
0 is chosen and let δ = . Then using the definition of limit we see that q 0 < (x1 − x01 )2 + (x2 − x02)2 < δ =  q ⇒ (x1 − x01 )2 <  √ ⇒ |(x1 − x01 )| < , a2 = |a| ⇒ |(f(x1 , x2) − x01 )| < ,

x1 = f(x1 , x2)

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That is |(f(x1 , x2) − x01 )| < 

whenever

0
0 3. C ONTINUITY A function f(x1,x2 ) is continuous at the point (x01 , x02 ) if 1. f is defined at (x01 , x02 ) 2. lim(x1 , x2 ) → (x01 , x02 ) f(x1 , x2) exists 3. lim(x1 , x2 ) → (x01 , x02 ) f(x1 , x2) = f(x01 , x02) The intuitive meaning of continuity is that if the point {x1 ,x2 } changes by a small amount, then the value of f(x1,x2 ) changes by a small amount. The means that the surface that is the graph of a continuous function has no hole or break. Sums, differences, products and quotients of continuous functions are continuous on their domains.

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4. D EFINITION

OF PARTIAL DIFFERENTIATION

Let f be a function with domain an open set in Rn and range in R1, i.e. y = f(x1 , x2 , ... , xn ). Now define the partial derivative of f with respect to xi as ∂f(x) f(x1 , x2, ...xi + h, ..., xn) − f(x1 , x2, ...xi, ..., xn) = fi (x) = lim h→ 0 ∂xi h

(3)

whenever the limit exists. The slope of the curve z = f(x1, x2 ) at the point (x01, x02 , f(x01 , x02 )) in the plane x2 = x02 is the value of the partial derivative of f with respect to x at (x01 , x02 ). 5. U SING

THE LIMIT CONCEPT TO COMPUTE A PARTIAL DERIVATIVE

5.1. Procedure. Add the vector h with zeros in all but one place to the vector x (h6=0) and compute f(x+h). Compute f(x) Compute the change in the value function: f(x+h) - f(x). For h 6= 0, form the quotient f (x + h)h − f (x) . Simplify the fraction in d as much as possible. Whenever possible, cancel h from both the numerator and denominator. f: f´(x) is the number that f (x + h)h − f (x) approaches as h tends to zero.

a: b: c: d: e:

5.2. Example. Let the function be f (x) = 2 x31 (x22 + 1) a: f (x + h ) = 2 (x1 + h)3 (x22 + 1) = 2 (x31 + 3 x21 h + 3 h2 x1 + h3) (x22 + 1) b: f (x) = 2 x31 (x22 + 1) c: f (x + h ) − f (x) = 2 ( 3 x21 h + 3 h2 x1 + h3 ) (x22 + 1) 2 ( 3 x21 h + 3 h2 x1 + h3 ) (x22 + 1) d: f (x + h)h − f (x) = h 2 ( 3 x2 h + 3 h2 x + h3 ) (x2 + 1)

1 1 2 e: = 2 ( 3 x21 + 3 h x1 + h2 ) (x22 + 1) h f: As h → 0, the expression goes to 6 x21 (x22 + 1) .

6. C ALCULATING

PARTIAL DERIVATIVES

6.1. The intuitive idea of computing a partial derivative. We calculate ∂∂xf1 by differentiating f with respect to x1 in the usual way while treating x2 as a constant. Similarly for other partial derivatives. 6.2. Example problems. a. f(x1 , x2) = x21 x2 + 8x21 x32 + x1 ln(x2 ) ∂z = 2x1 x2 + 16x1 x32 + ln(x2) ∂x1 ∂z x1 = x21 + 24x21 x22 + ∂x2 x2

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b.

15

f (x1, x2, x3 ) = x1 sin (x2 + 3x3) ∂f = sin (x2 + 3x3) ∂x1 ∂f = x1 cos (x2 + 3x3) ∂x2

c.

∂f = 3x1 cos(x2 + 3x3) ∂ x3 f (x1, x2 , x3 ) = 50 + 5x1 + 3 x2 + 2 x3 + 7x21 + 2 x1x2 + 3x1 x3 + 5 x22 + 4 x2 x3 + 2x23 ∂f = 5 + 14 x1 + 2 x2 + 3 x3 ∂x1 ∂f = 3 + 2 x1 + 10 x2 + 4 x3 ∂ x2 ∂f = 2 + 3 x1 + 4 x2 + 4 x3 ∂ x3 7. G EOMETRIC I NTERPRETATION

OF

PARTIAL D ERIVATIVES

2

When the domain of a function is R and the graph of the function is in R3 , a partial derivative with respect to one of the variables is the slope of a tangent line created when we intersect a vertical plane at a fixed level of the other variable with the surface R3 . Consider the function 1/4 1/2

f(x1 , x2) = 10x1 x2 which is shown in figure 21 along with a vertical plane at x2 = 10. Now consider figure 22 which highlights the intersection of the vertical plane and the surface repre1/4 √ senting the function. This line is a graph of the function f(x1 , 10) = 10x1 10. Figure 23 shows this intersection line alone. The partial derivative of f(x1,x2 ) with respect to x1 represents the slope of the tangent to this curve at a given point. Figure 24 shows the tangent to the curve representing the intersection of the vertical plane at x2 = 10 and the surface, while figure 25 shows all of the graphs together. Figure 26 shows the tangent to the curve representing the intersection of the vertical plane at x1 = 1 and the surface f(x1 , x2) = 12 (x1 − 1)x2 − (x1 − 1)2 − x22. Figure 27 shows the tangent to the curve representing the intersection of the vertical plane at x1 = 1 and the surface f(x1 , x2) = (x1 − 2) x2 + x22 + ex1 x2 . Figure 28 shows tangents to the curve in both the x1 and x2 directions for the surface f(x1 , x2) = 4e 8. TANGENT

LINES AND PLANES FOR FUNCTIONS WITH DOMAIN IN

2 −x2 1 + x2 4

+

x22 6 .

R2

The equation for the plane that is tangent to the graph of y = f(x1 ,x2 ) at the point {x01, x02 , f(x01 , x02 )} is

∂f(x01 , x02) ∂f(x01 , x02) x1 − x01 + x2 − x02 (4) ∂x1 ∂x2 Intuitively, this says we approximate the function f by its value at the point {x01 , x02 }, then move away ∂f (x01 , x02 ) from {x01 , x02 } along the plane that is tangent to the surface at that point. This plane has slope in ∂x1 y = f(x01 , x02) +

16

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1/4 1/2

F IGURE 21. Function f(x1 , x2) = 10x1 x2

with Vertical Plane at x2 = 10

20

x2 10 0 100 75 fHx1, x2L

50 25 0 0 10 x1

20

1/4 1/2

F IGURE 22. Intersection of function f(x1 , x2) = 10x1 x2

and Vertical Plane at x2 = 10

25 20

x2 15 10 5 100

0

80 fHx1, x2L

60 40 20 0 0 5 10 x1

15 20

∂f (x0 , x0 )

the x1 direction and slope ∂x1 2 2 in the x2 direction. Equation 4 is also called a second order Taylor series approximation to the function f at the point {x01 , x02 }.

A tangent plane contains the tangent lines when we hold x1 and x2 respectively constant. If we hold x2 constant at x02 , then equation 4 reduces to

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17

1/4

F IGURE 23. The function f(x1 , 10) = 10x1 101/2 25 20 15 10 5 100

0

80 60 40 20 0 0 5 10 15 20

1/4 1/2

F IGURE 24. Tangent to the function f(x1 , x2) = 10x1 x2 x2

when x2 is fixed at 10 and x1 = 4

20 15

10 5 0 80 fHx1 , x2 L 60 40 20 0 0 4 8 x1

y = f(x01 , x02) + =

f(x01 ,

x02)

12 16



∂f(x01 , x02) 0 ∂f(x01 , x02) x1 − x01 + x2 − x02 ∂x1 ∂x2


∂f(x01 , x02) x1 − x01 + ∂x1

which is just the equation for a tangent line when the domain of the function is R1 .

(5)

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1/4 1/2

F IGURE 25. Tangent to the function f(x1 , x2) = 10x1 x2

when x2 is fixed at 10 and x1 = 4

25 x2

20 15

10 5 100

0

80 fHx1, x2L

60 40 20 0 0 5 10 x1

15 20

Consider figure 24 where we hold x2 constant at 10. This looks just like a tangent graph when the domain of the function is the real line. Then consider figure 29 where we hold x1 constant at 4 and vary x2 . If we simplify this graph as with figure 24, we obtain a graph similar to a tangent graph when the domain of the function is the real line and the independent variable is x2 . This is shown in figure 30. If we rotate this graph as in figure 31, we can see clearly that it represents the value of f(x1 , x2) as a function of x2 . 1/4 1/2

The plane tangent to the surface f(x1 , x2) = 10x1 x2 at the point {x1 =4, x2 =10 } is shown in figure 32. The plane tangent to the surface f(x1 , x2) = (x1 − 2)x2 + x22 + ex1 x2 is shown in figure 33. 9. H IGHER

ORDER PARTIAL DERIVATIVES

9.1. Second order partial derivatives. When we differentiate a function f(x1 , x2) twice, we produce its second order derivatives. These derivatives are usually denoted by ∂ 2f ∂ x21 ∂ 2f ∂ x22 ∂ 2f ∂ x1 ∂ x2 ∂ 2f ∂ x2 ∂ x1

or

fx1 x1

or

f11

or

fx2 x2

or

f22

or

fx1 x2

or

f12

or

fx2 x1

or

f21

(6)

The defining equations are ∂2f ∂f ∂ = ∂x21 ∂ x1 ∂ x1 ∂ 2f ∂ ∂f = ∂x1∂x2 ∂x1 ∂ x2 Notice that the order in which the derivatives are taken, x2 , then with respect to x1 .

∂2f ∂ x1 ,∂ x2

(7)

means differentiate first with respect to

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F IGURE 26. Tangent to the function f(x1 , x2) = 12 (x1 − 1)x2 − (x1 − 1)2 − x22 when x1 is fixed at 1 and x2 = - 12

0

0.5 x1 1 1.5 2 1 0 -1 -2 -3 0.5

1

0 -0.5 -1

x1

Here are some example of first and second order partial derivatives.

a. Here is a function and the first and second order partial derivatives.

fHx1 , x2 L

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F IGURE 27. Tangent to the function f(x1 , x2) = (x1 − 2)x2 + x22 + ex1 x2 when x1 is fixed at 1 and x2 = 12

1.5 x1

1.25

1 0.75 40 30 fHx1 ,x2 L 20

0.5

10 0 -2

0

2 x2

4

f (x, w) = 50x2w + 3wx ∂f = 100xw + 3w ∂x ∂f = 50x2 + 3x ∂w ∂2f = 100w ∂x2 ∂2f = 100x + 3 ∂w∂x ∂2f = 100x + 3 ∂x ∂w ∂2f =0 ∂ w2

b. Here is a function and the first and second order partial derivatives.

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F IGURE 28. Tangents to the function f(x1 , x2) = 4e

0

2 −x2 1 + x2 4

21

+

x22 6

at {0.5, 0.3}

-2

2 6

4

2

0 -2

0

2

f (x1 x2, x3) = 50 + 5x1 + 3 x2 + 2 x3 + 7x21 + 2 x1x2 + 3x1 x3 + 5 x22 + 4 x2 x3 + 2x23 ∂f = 5 + 14x1 + 2x2 + 3x3 ∂x1 ∂f = 3 + 2x1 + 10x2 + 4x3 ∂ x2 ∂f = 2 + 3x1 + 4x2 + 4x3 ∂ x3 ∂2f = 14, ∂ x21

∂2f = 2, ∂ x2 ∂x1

∂2f = 2, ∂ x1 ∂x2

∂ 2f = 10, ∂ x22

∂2f = 3, ∂ x1 ∂ x3

∂2f = 4, ∂ x2 ∂x3

∂2 f = 3 ∂ x3 ∂ x1

∂2f = 4 ∂ x3 ∂ x2 ∂2f = 4 ∂ x23

22

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1/4 1/2

F IGURE 29. Tangent to the function f(x1 , x2) = 10x1 x2

when x1 is fixed at 4 and x2 = 10

25 x2

20 15

10 5 100

0

80 fHx1, x2L

60 40 20 0 0 5 10 x1

15 20

√ 1/2 F IGURE 30. Tangent to the function f(4, x2 ) = 10 2x2 when x2 = 10 x2

20 15

10 5 0 80 fHx1 , x2 L

60 40 20 0 0 4 8 x1

12 16

c. Here is another function and the first and second order partial derivatives.

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√ 1/2 F IGURE 31. Rotated graph of tangent to the function f(4, x2 ) = 10 2x2 when x2 = 10 8

x1

0 4

12 16

80 60 fHx1 , x2 L

40 20 0 0

5

10

15

20

x2

1/4 1/2

F IGURE 32. Plane tangent to the function f(x1 , x2) = 10x1 x2

at the point x1 = 4 and x2 = 10

20 x2 15 10 5 100

0

80 fHx1, x2L 60 40 20 0 0 5 10 x1

15 20

f(x1 , x2, x3) = 50x10.2x20.3x0.4 3 ∂f = 10x1−0.8x20.3x0.4 3 ∂x1 ∂f = 15x10.2x2−0.7x0.4 3 ∂x2 ∂f = 20x10.2x20.3x3−0.6 ∂x3 ∂2f = − 8x1−1.8x20.3x30.4, ∂x21

∂2f = 3x1−0.8x2−0.7x30.4, ∂x2∂x1

∂2f = 4x1−0.8x20.3x3−0.6 ∂x3∂x1

∂2f = 3x1−0.8x2−0.7x30.4, ∂x1∂x2

∂2f = −10.5x10.2x2−1.7x30.4, ∂x22

∂2f = 6x10.2x2−0.7x3−0.6 ∂x3∂x2

∂2f = 4x1−0.8x20.3x3−0.6, ∂x1∂x3

∂2f −0.7 −0.6 = 6x0.2 x3 , 1 x2 ∂x2 ∂x3

∂2f 0.3 −1.6 = −12x0.2 1 x2 x3 ∂x23

24

SIMPLE MULTIVARIATE CALCULUS

F IGURE 33. Plane tangent to the function f(x1 , x2) = (x1 − 2)x2 + x22 + ex1 x2 when x1 = 1 and x2 = 12 1.5 x1

1.25

1 0.75 0.5 15 fHx1,x2L 10 5 0 -2 0

2 x2

4

2

2

f f 9.2. Young’s theorem. As should be obvious from the examples, ∂x∂2 ∂x = ∂x∂1 ∂x . This is more generally 1 2 stated as Young’s theorem. If f(x1 , x2 , ... , xn ) and its partial derivatives f1 , f2, ... f11, f12, are defined throughout an open region containing a point ( x01, x02, x03, . . . , x0n ) and are all continuous at ( x01, x02, x03 , . . . , x0n ) 2 2 f f then ∂x∂j ∂x = ∂x∂i ∂x . i j

9.3. Higher order derivatives. We can compute higher order partial derivatives just as we computed higher order derivatives by simply differentiating again. Here is an example. Consider the function 1/4 1/2

f(x1 , x2) = 10x1 x2 The first order partial derivatives are ∂f 5 −3/4 1/2 = x1 x2 ∂x1 2 ∂f 1/4 −1/2 = 5x1 x2 ∂x2 The second order partial derivatives are

∂2f 15 −7/4 1/2 = − x1 x2 ∂x21 8 ∂2f 5 −3/4 −1/2 = x1 x2 ∂x2∂x1 4 ∂2f 5 −3/4 −1/2 = x1 x2 ∂x1∂x2 4 ∂2f 5 1/4 −3/2 = − x1 x2 ∂x22 2 The third order partial derivatives are

SIMPLE MULTIVARIATE CALCULUS

25

∂ 2f 105 −11/4 1/2 = x2 x 3 ∂x1 32 1 15 −7/4 −1/2 ∂2f = − x1 x2 ∂x2 ∂x1 ∂x1 16 ∂2f 15 −7/4 −1/2 = − x1 x2 ∂x1 ∂x2 ∂x1 16 ∂2f 5 −3/4 −3/2 = − x1 x2 ∂x2 ∂x2 ∂x1 8 ∂2f 5 −3/4 −3/2 = − x1 x2 ∂x1 ∂x2 ∂x2 8 ∂ 2f 15 1/4 −5/2 = x1 x2 ∂x32 4 10. D EFINITION

OF THE GRADIENT AND

H ESSIAN

OF A FUNCTION OF

n VARIABLES

10.1. Gradient of f. The gradient of a function of n variables f(x1 , x2, . . . , xn) is defined as follows. ∇ f(x) =



∂f ∂f ∂f ... ∂x1 ∂x2 ∂xn



∂f ∂x1 ∂f ∂x2

0



(8)

      =  .   ..    ∂f ∂xn

10.2. Hessian matrix of f. The Hessian matrix of a function of n variables f(x1 , x2, . . . , xn) is as is the n x n matrix of second order partial derivatives. It is symmetric due to Young’s theorem and looks as follows  2  ∂ f ∇2 f(x) = i, j = 1, 2, . . . , n ∂xi ∂xj 

   =    

∂ 2 f (x0 ) ∂x1 ∂x1 ∂ 2 f (x0 ) ∂x2 ∂x1

∂ 2 f (x0 ) ∂x1 ∂x2 ∂ 2 f (x0 ) ∂x2 ∂x2

.. .

.. .

... .. .

∂ 2 f (x0 ) ∂xn ∂x1

∂ 2 f (x0 ) ∂xn ∂x2

...

11. E VALUATING

THE

...



∂ 2 f (x0 ) ∂x1 ∂xn  ∂ 2 f (x0 )  ∂x2 ∂xn 

.. .

∂ 2 f (x0 ) ∂xn ∂xn

(9)

   

D OUBLE I NTEGRAL

11.1. definitions. Consider a closed and bounded set in