maths : Differentials and Partial Derivatives: Summary

**SUMMARY**

• Let *f* : (*a*,
*b*) → ℝ be a differentiable function and *x*_{0} ∈ (*a*, *b*) then linear
approximation *L* of *f* at *x*_{0}
is given by

L(*x*) = *f*
(*x*_{0}) + *f* ′ (*x*_{0})
(*x - x*_{0}) ∀ *x** *∈ (*a,b*)

• Absolute
error = Actual value – Approximate value

Relative
error = Absolute error / Actual value

Percentage
error =
Relative error × 100

(or)

Acutal
value / Absolute error × 100

• Let *f* : ( *a*
, *b*)→ ℝ be a differentiable function. For *x* ∈ ( *a*,*b*) and Δ*x* the
increment given to *x*, the
differential of *f* is defined by *df* = *f*
′(
*x* ) Δ*x *.

• All
the rules for limits (limit theorems) for functions of one variable also hold
true for

functions
of several variables.

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

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

exists
and it is denoted by

F has a
partial derivative with respect to* y *at
( x_{0} , y_{0} )∈ A

exists
and limit value is defined by

• Clariant’s
Theorem: Suppose that *A* = {(* x *,*
y *) a <* x *< b, c <* y *< d}⊂ ℝ^{2} , F : A → ℝ . If *f*_{xy} and* f *_{yx}
exist in *A* and are continuous in *A*, then* f _{ }*

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

• 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 dF = ∂F/∂*x*
*dx* + ∂F/∂*y* *dy* where* *Δ*x
*= *dx* and* *Δ*y *= *dy* .

• Suppose
*w* is a function of two variables *x*, *y*
where *x* and *y* are functions of a single variable ‘*t*’ then *d*w/*dt* = ∂*w/*∂*x*
⋅ *dxdt*
+ ∂*w/*∂*y* ⋅* dy/dt*

• Suppose
w is a function of two variables* x *and* y *where* x *and* y *are functions of
two variables s and t then,

• Suppose
that A = {(* x *,* y *) | a <* x *< b, c
<* y *< d}⊂ ℝ^{2} , F : A → ℝ^{2} . If F is having continuous partial
derivatives and homogeneous on A, with degree p, then* x *∂F/∂*x* +* y *∂F/∂*y* = pF.

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

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12th Maths : UNIT 8 : Differentials and Partial Derivatives : Summary | Differentials and Partial Derivatives | Mathematics

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