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

Chapter: 12th Maths : UNIT 8 : Differentials and Partial Derivatives

Summary

maths : Differentials and Partial Derivatives: Summary

SUMMARY

• Let f : (a, b) be a differentiable function and x0 (a, b) then linear approximation L of f at x0 is given by

L(x) = f (x0) + f  ′ (x0) (x - x0 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)→ be a differentiable function. For x ( a,b) and Δx the increment given to x, the differential of f is defined by df = f ′( x ) Δx .

• 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 } 2 , F : A and ( x0 , y0 ) A .

• (i) F has a partial derivative with respect to x at ( x0 , y0 ) A


exists and it is denoted by 

F has a partial derivative with respect to y at ( x0 , y0 )

exists and limit value is defined by 

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

• Let A = {( x , y ) | a < x < b, c < y < d} 2. A function U : A → is said to be harmonic in A if it satisfies ∂2u/∂x2 + ∂2u/y2 = 0, ( x , y) A . This equation is called Laplace’s equation.

• Let A = {( x , y ) | a < x < b, c < y < d} 2 , F : A and ( x0 , y0 ) A .

(i) The linear approximation of F at ( x0 , y0 ) A is defined to be


(ii) The differential of F is defined to be dF = ∂F/∂x dx + ∂F/∂y dy where Δx = dx and Δy = dy .

• Suppose w is a function of two variables x, y where x and y are functions of a single variable ‘t’ then  dw/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} 2 , F : A → 2 . If F is having continuous partial derivatives and homogeneous on A, with degree p, then x ∂F/∂x + y ∂F/∂y = pF.

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12th Maths : UNIT 8 : Differentials and Partial Derivatives


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