Here we introduce similar ideas for functions of two variables and three variables. In general for functions of several variables these concepts can be defined similarly.

**Linear
Approximation and Differential of a function of several variables**

Earlier
in this chapter, we have seen that linear approximation and differential of a
function of one variable. Here we introduce similar ideas for functions of two
variables and three variables. In general for functions of several variables
these concepts can be defined similarly.

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

(i) The linear approximation of F at ( *x*_{0} , *y*_{0}
) ∈ *A* is defined to be

(ii) The differential of F is defined to be

where *dx* = Δ*x *and*
dy *=* *Δ*y*,

Here we
shall outline the linear approximations and differential for the functions of
three variables. Actually, we can define linear approximations and differential
for real valued function having more variables, but we restrict ourselves to
only three variables.

Let A = {( x,* y *, z) **|** *a*
<* x *< *b*, *c* <* *y*
*< *d*, *e* < *z* <* f *} ⊂ ℝ^{3}, F : A → ℝ and ( *x*_{0} , *y*_{0} , *z*_{0}
) ∈ A .

(i) The linear approximation of F at ( *x*_{0} , *y*_{0}
, *z*_{0} ) ∈ A is defined to be

(ii) The differential of F is defined by

where *dx* = Δ*x *,*
dy *=* *Δ*y *and d*z* = Δ*z* ,

Geometrically,
in the case of function *f* of one
variable, the linear approximation at a point *x*_{0} represents the tangent line to the graph of *y* = *f*
( *x*) at *x*_{0}* *.
Similarly, in the case of a function* F *of
two* *variables, the linear
approximation at a point ( *x*_{0}
, *y*_{0} ) represents the
tangent plane to the graph of *z* =
*F* ( *x* , *y*) at ( *x*_{0} , *y*_{0} ) .

** **

**Example 8.16**

If *w*( *x*,
*y* , *z*) = *x*^{2}*y* + *y*^{2}*z* + *z*^{2}*x*, *x*
, *y*, *z* ∈ ℝ, find the differential *dw*.

**Solution**

First
let us find *w _{x}* ,

Now *w*_{x} = 2x*y +* z^{2}, w* _{y}
*= 2

Thus,by
(15), the differential is

*dw *=* *(2*xy *+* z*^{2}* *)*dx *+* *(2* yz *+* x*^{2}* *)*dy *+* *(2*zx *+* y*^{2})*dz
*.

** **

**Example 8.17**

Let *U *(*
x*,* y *,* z*)* *=* x *^{2}* *−* xy *+* *3sin* z*,* x *,* y*,*
z *∈* *ℝ. Find the linear approximation for *U *at (2, −1, 0) .

**Solution**

By (14),
linear approximation is given by

Now *U _{x }*= 2

Here (*x*_{0} , *y*_{0} , *z*_{0}
) =
(2, −1,
0) , hence *U _{x}* (2, −1,
0) =
5,

Thus *L *(*
x*,* y *,* z*)* *=* *6* *+* *5(*x *−* *2)* *−* *2(* y *+* *1)* *+* *3(*z *−* *0)* *=* *5*x *−* *2* y *+* *3*z *−* *6* *is the required

Linear approximation
for *U* at (2, −1,
0) .

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

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12th Maths : UNIT 8 : Differentials and Partial Derivatives : Linear Approximation and Differential of a function of several variables | Mathematics

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