a)Sketch some level curves of the function f(x,y)=(x^2+y^2)/x b) Verify Clairaut's theorem for f(x,y) = arctan (y/x). ` `

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aruv | High School Teacher | (Level 2) Valedictorian

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Verify Clairaut's theorem for f(x,y) = arctan (y/x).

A function f is defined at a point (a,b) in its domain of definition. If

`f_(xy) and f_(yx)` are continuous in domain of definition of f, then

`f_(xy)(a,b)=f_(yx)(a,b)`

Given

`f(x,y)=tan^(-1)(y/x)`   (i)

Differentiate (i) partially  with respect to x and y respectively.

`(delf(x,y))/(delx )=f_x=1/(1+(y/x)^2)del/(delx)(y/x)`

`=(x^2/(x^2+y^2))(-y/x^2)`

`=-y/(x^2+y^2)`

`(delf_x)/(del y)=(del(-y/(x^2+y^2)))/(dely)`

`f_(xy)=-((x^2+y^2)-y(2y))/(x^2+y^2)^2`

`f_(xy)=-(x^2-y^2)/(x^2+y^2)^2`

`(delf(x,y))/(dely)=f_y=1/(1+(y/x)^2)(del(y/x))/(dely)`

`=(x^2/(x^2+y^2))(1/x)=x/(x^2+y^2)`

`(delf_y)/(delx)=(del(x/(x^2+y^2)))/(delx)`

`=((x^2+y^2)-x(2x))/(x^2+y^2)^2`

`=(y^2-x^2)/(x^2+y^2)^2`

`=-(x^2-y^2)/(x^2+y^2)^2`

`Thus`

`f_(xy)(x,y)=f_(yx)(x,y)`

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