For each this is a system of n linear equations in which the derivatives (1 sis\
ID: 3109819 • Letter: F
Question
For each this is a system of n linear equations in which the derivatives (1 sis") are the ion Theorem 9.29 Example n 2, m 3, and consider the mapping f fi, f) of Rs into R2 given by 4y2 3 x2, y1, y2, y3) 6x1 2y, y3 x2 cos x With respect to the standard bases, the matrix of the transformation A f'(a, b) is 2 3 1 4 01 [A] -6 1 2 0 -1 Hence [Ax] LAy] 2 0 -1 -6 We see that the column vectors of [Ar] are independent. Hence A is invertible and the implicit function theorem asserts the existence of a 8'-mapping g, defined in a neighborhood of (3, 2, 7), such that g(3, 2, 7) (0, 1) and f(g(y), y) 30. We can use (58) to compute g 2, 7): Since [1 -3 6 20 gives (58) ri [1 -4 01 t t -To lg'(3, 2, 7) 20 L6 2 0 -1 t To.Explanation / Answer
n=2 means 2 variable of x here x1 and x2
m=3 means 3 variable of y here y1, y2 and y3
a(0,1) i.e x1=0, x2= 1
b(3,2,7 ) i.e y1=3 , y2=2 ,y3=7
now substituting all value of variable in f1 and f2
f1(a,b)=2+3-8+3=0
f2(a,b)=1-0+6-7=0
f( f1,f2) =(0,0) =0
Now
A=f'(a,b)= [ f1'(a,b) ; f2'(a,b ) ]
thats is take derivatives and substitute value of ( a,b) i.e all variable value
row 1 corresponding to f1 and row 2 corresponding to f2
column represent which is variable while taking derivatives
example
A [row 1 and column 2 means ] take derivative of f1 w.r.t x2 and substitute (a,b)
f1'x2=y1 = 3
A [row 2 and column 2 means ] take derivative of f2 w.r.t x2 and substitute (a,b)
f2'x2=cosx1=cos0=1
now Ax is invertible if deteminant of Ax is not zero
determinant is not zero only when the column of Ax is independent
Here independent means relation column 1 = k * column 2 is not possible for some constant value of k
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