its a power point presentation of partial derivative equation

1. CREATED BY :-
UTKARSH PATEL (170010114038)
SUBMITTED TO :- NBD SIR

2. Partial Derivative Equations

3.  The partial derivative of f(x,y) w.r.t. x at a point
 The partial derivative of f(x,y) w.r.t. y at a point
(1)h
yxfyhxf
x
f
f
h
x yx
),(),(
lim
0000
00
0
)( ,
−+
=





∂
∂
=
→
k
yxfkyxf
y
f
y k
yxf ),(),(
lim
000
00
0
0
)( ,
−+
=





∂
∂
=
→
)( 00, yx
)( 00, yx

4. xyyxyx ffff ,,,• If (x,y) and its partial derivative
are defined throughout an open region and containing
a point(a,b) and all the are continuous at point (a,b)
then (a,b) then (a,b)
• Which is also known as clairaut’s theorem.
xyf yxf
f

5. • Laplace equation in 2-D is 02
2
2
2
=
∂
∂
+
∂
∂
y
f
x
f
• Laplace equation in 3-D is 02
2
2
2
2
2
=
∂
∂
+
∂
∂
+
∂
∂
z
f
y
f
x
f

6. 0
6,6
3
2
2
2
2
2
2
2
2
23
=
∂
∂
+
∂
∂
−=
∂
∂
=
∂
∂
−=
y
u
x
u
x
y
u
x
x
u
xyxu

7. xy
z
yx
z
∂∂
∂
=
∂∂
∂ 22
• If having continuous second order
partial derivatives then both are
equal.
• This also commutative property.
),( yxfu =

8. )(
...
x
x
yx
yy
z
SHL
yxz
+
∂
∂
=
∂
∂
+=
1−
= x
xy
( )1
. −
∂
∂
=





∂
∂
∂
∂ x
yx
xy
z
x
)log( 11
yyxy xx −−
+=
[ ]yxy
yx
z
SHL
x
log1
...
1
2
+=
∂∂
∂ −
…..(1)
Again
( )x
yx
xx
z
SHR
+
∂
∂
=
∂
∂
...
( )yy
yy
z
y
x
log1+
∂
∂
=





∂
∂
∂
∂
11
1
log
log
−−
−
+=
+=
xx
x
x
yyxy
y
y
yxy
[ ] )2.......(log1
...
1
2
yxy
xy
z
SHR
x
+=
∂∂
∂ −
Form (1) & (2) clairaut’s
Equation is clerified

9.  For two independent value of functions
 w= has continuous partial derivative
and if x=x(t) , y=y(t) are differentiable
function of t than the composite function
is a differentiable function of t and derivative is
),( yxf
yx f
y
f
f
x
f
=
∂
∂
=
∂
∂
,
))(),(( tytxfW =
t
y
y
w
t
x
x
w
dt
dw
∂
∂
∂
∂
+
∂
∂
∂
∂
=

10. tytx
xyyxu
cot,2sin
42
==
+=
Than find
dt
du
when 0,00 ==⇒= yxt
Now
t
y
y
u
t
x
x
u
dt
du
∂
∂
∂
∂
+
∂
∂
∂
∂
=
324
12,32 xyx
dy
du
yxy
dx
du
+=+=
t
dt
dx
t
dt
dx
sin,2cos2 −==
Than
So ( )( ) ( )( )txyxtyxy
dt
du
sin12cos232 324
−+++=
txytxtytxy
dt
du
sin12sin2cos62cos4 324
−−+=
6
)0sin()1)(0(12)0sin()0()0cos()1(6)0cos()1)(0(4
0
324
=
−−+=
=
dt
du
dt
du
tNow

11.  The equation of the form =c is known as an
implicit function.
 If x is a function of two variable & y is again
function of x. Therefore we may regard is a
composite function x.
 Therefore derivative of w.r.t. x is
),( yxf
f
f
0,
)1(0
≠−=
∂
∂
∂
∂
+
∂
∂
=
∂
∂
∂
∂
+
∂
∂
∂
∂
=
y
y
x
f
f
f
dx
dy
x
y
y
f
x
f
x
y
y
f
x
x
x
f
dx
df

12. xay
xxa
x
y
f
f
x
y
xayf
yaxf
xyayxyxf
xyayx
y
x
y
x
23
32
23
23
244
244
54
45
054
054
05),(
5
−
−
=
∂
∂
−=
∂
∂
=−=
=−=
=−+=
=+

13.  If is a function of two independent variable x &
y is said to be homogeneous function of degree n.
 If it satisfied where is constant.
 E.g.

 It is Homogeneous Function
),( yxf
),(),( yxfyxf n
λλλ = λ
)2(),(
2),(
)2()()(),(
2),(
2333
223333
233
233
yxyxyxf
yxyxyxf
yxyxyxf
yxyxyxfu
++=
++=
++=
++==
λλλ
λλλλλλ
λλλλλ