Kenneth Kuttler

454 CHAPTER 21. FUNCTIONS OF MANY VARIABLES

(a) x2y2z+w(b) e2 + xy+ z2

(d) ln(x2 + y2 +1

)+ ez

(e) sin(xyz)+ cos(xy)

11. Find ∂ f∂x ,

∂ f∂y , and ∂ f

∂ z for f =

(a) x2y+ cos(xy)+ z3y

(b) ex2+y2zsin(x+ y)

ex2+y3)

(d) x2 cos(sin(tan(z2 + y2

)))(e) xy2+z

12. Suppose

f (x,y) =

{2xy+6x3+12xy2+18yx2+36y3+sin(x3)+tan(3y3)

3x2+6y2 if (x,y) ̸= (0,0)0 if (x,y) = (0,0) .

Find ∂ f∂x (0,0) and ∂ f

∂y (0,0).

13. Why must the vector in the definition of the directional derivative be a unit vector?Hint: Suppose not. Would the directional derivative be a correct manifestation ofsteepness?

21.6 Mixed Partial DerivativesUnder certain conditions the mixed partial derivatives will always be equal. This aston-ishing fact may have been known to Euler in 1734.2

Theorem 21.6.1 Suppose f : U ⊆R2 →R where U is an open set on which fx, fy,fxy and fyx exist. Then if fxy and fyx are continuous at the point (x,y) ∈U, it follows

fxy (x,y) = fyx (x,y) .

Proof: Since U is open, there exists r > 0 such that B((x,y) ,r)⊆U . Now let |t| , |s|<r/2 and consider

∆(s, t)≡ 1st{

h(t)︷︸︸︷f (x+ t,y+ s)− f (x+ t,y)−

h(0)︷︸︸︷( f (x,y+ s)− f (x,y))}. (21.6)

2Leonhard Euler 15 April 1707 - 18 September 1783 was the most prolific mathematician ever to have lived.His contributions also included fundamental work in fluid mechanics and engineering. For example, the formulafor the stiffness of a beam which involves a moment of inertia is due to him. He wrote about 30,000 pages. Heeven wrote on music and theology. With Lagrange, he invented calculus of variations in which one looks for anunknown function maximizing a functional.

Euler had the ability to do huge computations in his head. He also had a memory which allowed him tomemorize entire works of literature such as the Aeneid. He is also remembered for his work in logic, numbertheory, and graph theory. The notation π and e are due to him as is Euler’s formula discussed earlier.

He was a kind and generous man and a devout Christian who believed the Bible was inspired. For the last partof his life, he was essentially blind. They didn’t know how to treat things like cataracts back then.

454 CHAPTER 21. FUNCTIONS OF MANY VARIABLES(a) x’y?ztw (d) In(x+y* +1) +e(b) e?+xy+27(c) sin (z*) +-cos (xy) (e) sin (xyz) + cos (xy)4 Of a QLL. Find $f, $4, and Sf for f =(a) x*y +cos (xy) +2y (d) x*cos (sin (tan (z? + y*)))(b) e* +”zsin(x+y)(c) 2 sin? (e"»") (e) +12. SupposeQnxy+6x3 + L2xy? + 18yx2+36y>+sin x? )+tan 3y3 .fey) = won inl) sen) it (x,y) A (0,0)O if (x,y) = (0,0).: a ofFind $f (0,0) and $f (0,0).13. Why must the vector in the definition of the directional derivative be a unit vector?Hint: Suppose not. Would the directional derivative be a correct manifestation ofsteepness?21.6 Mixed Partial DerivativesUnder certain conditions the mixed partial derivatives will always be equal. This aston-ishing fact may have been known to Euler in 1734.7Theorem 21.6.1 Suppose f :U CR? +R where U is an open set on which fos fyfry and fy exist. Then if fry and fy, are continuous at the point (x,y) € U, it followsSay (x,y) = fix (x,y) .Proof: Since U is open, there exists r > 0 such that B((x,y),r) C U. Now let ||, |s| <r/2 and considerA(t) h(0)A(s.t)=—{fFlettyts)— Fatty) —-Fayts)—fayy} G16)?Leonhard Euler 15 April 1707 - 18 September 1783 was the most prolific mathematician ever to have lived.His contributions also included fundamental work in fluid mechanics and engineering. For example, the formulafor the stiffness of a beam which involves a moment of inertia is due to him. He wrote about 30,000 pages. Heeven wrote on music and theology. With Lagrange, he invented calculus of variations in which one looks for anunknown function maximizing a functional.Euler had the ability to do huge computations in his head. He also had a memory which allowed him tomemorize entire works of literature such as the Aeneid. He is also remembered for his work in logic, numbertheory, and graph theory. The notation z and e are due to him as is Euler’s formula discussed earlier.He was a kind and generous man and a devout Christian who believed the Bible was inspired. For the last partof his life, he was essentially blind. They didn’t know how to treat things like cataracts back then.