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DERIVATIVES IN MULTI
PORAMATE (TOM) PRANAYANUNTANA
In Calc 1:
For f : D ⊂ R→ R ⊂ R,(1)
f ′(a) := limx→a
f(x)− f(a)
x− a.(2)
A function f is differentiable at a if, upon continued magnification of the graph about thepoint (a, f(a)), the graph is indistinguishable from a straight line; that is, f is differentiableat a if
limx→a
f(x)− f(a)− f ′(a)(x− a)
x− a= 0.(3)
This can be generalized to Multi as follows:
In Multi:
For f : D ⊂ R2 → R ⊂ R,(4)
A function of 2 variables, f(x, y), is differentiable at a point (a, b) if the graph of z = f(x, y)near that point (a, b) is indistinguishable from a plane; that is
lim(x, y)︸ ︷︷ ︸
~r
→(a, b)︸ ︷︷ ︸~a
|f(x, y)− f(a, b)−
T (
xy
− ab
)︷ ︸︸ ︷T (~r − ~a) |
‖~r − ~a‖= 0,(5)
Date: June 20, 2015.
Derivatives in Multi Poramate (Tom) Pranayanuntana
or, from an equation of the tangent plane z = L(x, y) = f(a, b)+fx(a, b)(x−a)+fy(a, b)(y−b)(if it uniquely exists), we have f(x, y) is differentiable at a point (a, b) if
lim(x, y)︸ ︷︷ ︸
~r
→(a, b)︸ ︷︷ ︸~a
|f(x, y)−L(x,y)︷ ︸︸ ︷
(f(a, b) + fx(a, b)(x− a) + fy(a, b)(y − b)) |∥∥∥∥[ xy]−[ab
]∥∥∥∥ = 0.(6)
L(x, y) = f(a, b) +[fx(a, b) fy(a, b)
]︸ ︷︷ ︸Jf(~a)=Jf(a,b)
[x− ay − b
]︸ ︷︷ ︸
T (~r−~a)
(7)
= f(a, b) +
[fx(a, b)fy(a, b)
]︸ ︷︷ ︸
grad f(a,b)=∇f(a,b)
�
[x− ay − b
] ,
where ∇(◦) =
[∂(◦)/∂x∂(◦)/∂y
]=
[(◦)x(◦)y
]is a vector derivative operator.
Compare the following:
Calc 1 Multi
limx→a
|f(x)−l(x): tangent line of f at a︷ ︸︸ ︷
(f(a) + f ′(a)(x− a)) ||x− a|
= 0 lim~r→~a
|f(~r)−L(~r): tangent plane of f at ~a︷ ︸︸ ︷(f(~a) + Jf(~a)(~r − ~a)) |‖~r − ~a‖
= 0
We can see that Jf(~a) is the derivative of z = f(~r) = f(x, y) at ~r = ~a = (a, b).
In this class, we will use dot product instead of matrix multiplication, so the derivative
matrix Jf(~a) will be represented by the gradient vector, grad f(a, b) =
[fx(a, b)fy(a, b)
], in the
xy-plane, which the domain of f is part of.
June 20, 2015 Page 2 of 5
Derivatives in Multi Poramate (Tom) Pranayanuntana
Directional Derivatives
Calc 1
∆y = f(x)− f(a) ≈ f ′(a)(x− a) = f ′(a)∆x
(8)
= (f ′(a) · u) |∆x|= Duf(a) |∆x| ,
where u is the unit vector pointing in the di-
rection of ∆x = x− a.
Multi
∆z = f(~r)− f(~a) ≈ Jf(~a)(~r − ~a)(9)
= (grad f(a, b) � (~r − ~a))
= (grad f(a, b) � u) ‖~r − ~a‖= Duf(a, b) ‖~r − ~a‖ ,
where u is the unit vector pointing in thedirection of ∆~r = ~r − ~a.
The directional derivative of f(x, y) at (a, b) in the direction of u =∆~r
‖∆~r‖in the domain of f
in the xy-plane, denoted by Duf(a, b), is limRun→0
Rise
Run= lim‖∆~r‖→0
∆z
‖∆~r‖=
(∇f(a, b) �
∆~r
‖∆~r‖
)=
(∇f(a, b) � u).
June 20, 2015 Page 3 of 5
Derivatives in Multi Poramate (Tom) Pranayanuntana
It can also be seen that
Duf(a, b) := limRun=h→0
Rise︷ ︸︸ ︷f(a+hu1,b+hu2)︷ ︸︸ ︷
f(
[ab
]+ h
[u1
u2
])−f(a, b)
h︸︷︷︸Run
(10)
= limh6=0,h→0
f(
x︷ ︸︸ ︷a+ hu1,
y︷ ︸︸ ︷b+ hu2)− f(a, b)
h;
with f(x, y)− f(a, b) ≈ fx(a, b)(x− a) + fy(a, b)(y − b)
= limh6=0,h→0
fx(a, b)(��hu1) + fy(a, b)(��hu2)
��h
=
([fx(a, b)fy(a, b)
]�
[u1
u2
])= (∇f(a, b) � u)
= ‖∇f(a, b)‖ ‖u‖ cos θ, 0 ≤ θ ≤ π.
June 20, 2015 Page 4 of 5
Derivatives in Multi Poramate (Tom) Pranayanuntana
From Duf(a, b) = ‖∇f(a, b)‖ ‖u‖ cos θ, 0 ≤ θ ≤ π. Since ‖u‖ = 1 and ‖∇f(a, b)‖ is a fixedpositive number (once point (a, b) is picked), therefore
maxDuf(a, b) = ‖∇f(a, b)‖, when cos θ = 1 that is when θ = 0 or when u points in directionof ∇f(a, b).
∆z = f(~r)− f(~a) ≈ Jf(~a)(~r − ~a)
= (grad f(a, b) � (~r − ~a))
= (grad f(a, b) � u) ‖~r − ~a‖= Duf(a, b) ‖~r − ~a‖ .
Properties of grad f(a, b) = ∇f(a, b)
If ∇f(a, b) 6= ~0. Then
Direction Properties
• ∇f(a, b) points in direction of maximum rate of increasing of f(a, b),
• Direction of ∇f(a, b) is perpendicular to the contour line z = f(a, b)
(in the domain of f in the xy-plane)
Magnitude Properties
• maxDu=∇f(a,b)/‖∇f(a,b)‖f(a, b) = ‖∇f(a, b)‖���>
1‖u‖ ���:
1cos 0
That is ‖∇f(a, b)‖ = maxDuf(a, b) or
‖∇f(a, b)‖ = maximum rate of change of f at (a, b)
• ‖∇f(a, b)‖
� is large when contour lines (of fixed ∆z) of f are closer together.
� is small when contour lines (of fixed ∆z) of f are further apart.
June 20, 2015 Page 5 of 5
