In this section, we shall see how the concept of derivative for functions of one variable may be generalized to real-valued function of several variables. First we consider functions of two variables.

**Partial
Derivatives**

In this
section, we shall see how the concept of derivative for functions of one
variable may be generalized to real-valued function of several variables. First
we consider functions of two variables.

Let *A* = { ( *x*, *y* )** | ***a*
<
*x* < *b*,
*c* < *y*
<
*d *} ⊂ ℝ^{2} , and *F* : *A* →
ℝ be a real-valued function. Suppose
that (*x*_{0}* *,*
y*_{0}* *)* *∈* A *; and we are interested in finding
the rate of change of* F *at* *(*
x*_{0}* *,* y*_{0}* *)* *with respect to the* *change **only** in the variable *x* .
As we have seen above *F* ( *x*, *y*_{0}
) is a function of *x* alone and it
will represent a curve obtained by intersecting the surface *z* = *F*
( *x* , *y*) with *y* =
*y*_{0} plane. So we can
discuss the slope of the tangent to the curve *z* = *F* ( *x* , *y*_{0}
) at *x* = *x*_{0}
by finding derivative of *F* ( *x*, *y*_{0}
) with respect to *x* and evaluating it
at *x* = *x*_{0}
. Similarly, we can find the slope of the curve *z* = *F* ( *x*_{0} , *y*) at *y* =
*y*_{0} by finding derivative
of *F* ( *x*_{0} , *y*) with
respect to *y* and evaluating it at *y* = *y*_{0}
. These are the key ideas that motivate us to define partial derivatives below.

** **

Let *A* = {( *x*, *y* )**|** *a*
<
*x* < *b*,
*c* < *y*
<
*d* } ⊂ ℝ^{2} , *F* : *A* →
and ( *x*_{0} , *y*_{0} ) ∈ *A*

(i) We
say that *F* has a partial derivative
with respect to *x* at ( *x*_{0} , *y*_{0} ) ∈
*A* if

exists.
In this case, the limit value is denoted by ∂F/∂*x* ( *x*_{0} , *y*_{0} ) .

(ii) We
say F has a partial derivative with respect to* y *at ( *x*_{0} , *y*_{0} ) ∈ A if

exists.
In this case, the limit value is denoted by ∂F/∂*y* ( *x*_{0} , *y*_{0} ) .

**Remarks**

1. Partial
derivatives for functions of three or more variables are defined exactly in a
similar manner.

2. We
read ∂*F* as **“partial** *F*
**”** and ∂*x* as **“partial**
*x* **”**. And we read ∂*F/*∂*x* as “partial *F* by partial
*x* ”. It is also read as “dho *F* by dho *x* ”.

3. Similarly,
we read ∂F/∂*x* as “partial F by
partial* y *” or as “dho F by dho* y *.

4. Sometimes
∂F/∂*x* (* x _{0} *,

Similarly
∂F/∂*x* Fy (x_{0} , y_{0}
) is denoted by F_{y}(* x _{0}
*,

5. An
important thing to notice is that while finding partial derivative of *F* with respect to *x* , we treat the *y*
variable as a constant and find derivative with respect to *x* . That is, except for the variable with respect to which we find
partial derivative, all other variables are treated as constants. That is why
we call it as **“partial
derivative”**.

6. If *F* has a partial derivative with respect
to *x* at every point of *A* , then we say that ∂F/∂*x*^{ }( *x* , *y*)
exists on *A* . Note that in this case ∂F/∂*x* ( *x*
, *y*) is again a real-valued function
defined on *A* .

7. In
the light of (4) , it is easy to see that all the rules **(Sum, Product, Quotient, and Chain rules)**
of differentiation and formulae that we have learnt earlier hold true for the
partial differentiation also.

Recall
that for a function of one variable, differentiability at a point always
implies continuity at that point. For a function *F* of two variables *x* , *y* we have defined ∂F/∂x (*u* , *v*)
and ∂F/∂y (*u* , *v*) . Do the existence of partial derivatives of *F* at a point (*u* , *v*) implies continuity
of *F* at (*u* , *v*) ? Following
example illustrates that this may not necessarily happen always.

** **

**Example 8.11**

Let* f *( *x
, y*) = 0 if xy ≠ 0 and* f *( *x , y*) = 1 if xy = 0 .

(i) Calculate
: ∂*f* **/**∂x ( 0, 0), ∂*f* **/**∂x(0, 0).

(ii) Show
that* f *is not continuous at (0, 0) .

**Solution**

Note
that the function *f* takes value 1 on
the *x* , *y* -axes and 0 everywhere else on **R**^{2} . So let us calculate

This
completes (i).

Now for
(ii) let us calculate the limit of *f* as
( *x* , *y*) → (0, 0) along the line *y *=* x *. Then lim_{( x , y ) →(0 , 0)}* f *(*x
, y*) = 0 ; because along the line* y *=* x *when* x *≠ 0,* f *(* x *, y) = 0 , But* f *(0, 0) = 1 ≠ 0 ; hence *f*
cannot be continuous at (0, 0).

** **

**Example 8.12**

Let F ( *x, y *) = x^{3}*y *+ y^{2}*x *+ 7 for all (* x *, y) ∈ ℝ^{2} . Calculate ∂F/∂*x*(−1, 3) and ∂F/∂*x*(−2,1) .

**Solution**

First we
shall calculate ∂F/∂*x*(*x*, *y*),
then we evaluate it at (−1, 3) . As we have already

observed,
we find the derivative with respect to *x*
holding *y* as a constant. That is,

= 3x^{2}y
+ y^{2} + 0

= 3x^{2}y
+ y^{2}.

So ∂F/∂*x* (−1, 3) = 3(−1)^{2} 3 + 3^{2}
= 18 .

Next
similarly we find partial derivative with respect to y.

*= x*^{3}* *+* *2* yx *+* *0 *= x*^{3}* *+* *2* yx *. Hence we
have ∂F/∂*x*(−2,1)
=
(−2
)^{3} + 2(1)(−2 ) = −12 .

Note
that in the above example ∂F/∂*x*(*x* , *y*)
=
3*x*^{2}*y* + *y*^{2} ,
which is again a function of two variables. So we can take the partial
derivative of this function with respect to *x*
or *y* . For instance, if we take G (
x,* y *) = 3x^{2}* y *+*
y*^{2} , then we find ∂G/∂*x*
= 6xy . Since G ( *x , y*) = ∂F/∂*x* , we have ∂G/∂*x* = ∂//∂*x*(∂F/∂*x*) = 6xy.

We
denote this as ∂^{2}*F/*/∂^{2}*x* ;
which is called the second order partial derivative
of *F* with respect to *x*. ^{}

Also, ∂G/∂y
= 3x^{2} + 2* y *. Since G ( *x , y*) = ∂F/∂y , we have ∂G/∂y = ∂/∂y(∂F/∂y)
= 3*x*^{2} + 2*y *. We denote this as ∂^{2}F / ∂y∂x
; which is called the mixed partial derivative
of F with respect to* x *,* y *. Similarly we can also calculate ∂/∂x
(∂F/∂y) = 3x^{2} + 2*y *.

Also, if
we differentiate ∂F / ∂y (* x *, y) partially
with respect to* y *we obtain ∂^{2}F
/ ∂y^{2}; which is called the second order partial derivatives of F
with respect to* y *. So for any function
F defined on any subset {(*x,y*) | a
<* x *< b, c <* y *< d} ⊂ ℝ^{2} we have the following notation :

All the
above are called second order partial derivatives of F . Similarly we can
define higher order partial derivatives. For example,

Next we
shall see more examples on partial differentiation.

** **

**Example 8.13**

Let* f *(x,* y *) = sin(*xy*^{2}
) + e^{x3+5y}* *for all (* x *, y) ∈
ℝ^{2} . Calculate

**Solution**

Note
that we have first used sum rule, then in the next step we have used chain
rule. In the third step, product rule is used. Also, we see that *f* * _{xy}*
=

Suppose that *A* = {( *x*, *y* ) | *a* < *x* < *b*, *c* < *y* < *d*} ⊂ ℝ^{2}, *F* : *A* → ℝ. If *f _{xy}*
and

We omit the discussion on the proof at this stage.

** **

Let A = { (* x *,* y *) | *a < x b, c < y < d* } ⊂ ℝ^{2}. A function* u *: A → ℝ^{2} is said to be **harmonic** in A if it satisfies ∂^{2}*u/*∂^{2}*x* + ∂^{2}*u/*∂^{2}*y* = 0, ∀(*x , y*) ∈ A . This equation is
called **Laplace’s** equation.

Laplace’s
equation occurs in the study of many natural phenomena like heat conduction,
electrostatic field, fluid flows etc.

** **

Let *u* ( *x*,
*y* ) = *e*^{−}^{2y} cos(2*x*) for all (
*x* , *y*) ∈
ℝ^{2} . Prove that *u *is a harmonic function in ℝ^{2 }.

We need to
show that *u* satisfies the Laplace’s equation
in ℝ^{2 }. Observe that *u _{x}* (

Similarly,
*u* * _{y}*
(

Thus, *u* * _{xx}*
+

Tags : Mathematics , 12th Maths : UNIT 8 : Differentials and Partial Derivatives

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