Physical Significance of d(f(x,y))

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• April 12th 2013, 06:55 AM
rickrishav
Physical Significance of d(f(x,y))
What is the physical meaning of d(x2y2)?

For example,
d(x2) = 2x dx
This can be derived as,

d(x2)/dx = 2x
d(x2) = 2x dx

Here dx can be thought of Delta x, rather than thinking d/dx as an operation. So, small change in x2 is the product of "two times current value of x" and small change in x.

What will be the similar deduction methodology for d(x2y2) = 2xy2dx + 2yx2dy?

Thanks a lot.
• April 12th 2013, 09:16 AM
HallsofIvy
Re: Physical Significance of d(f(x,y))
Given $f(x,y)= x^2y^2$ then $grad f= \nabla f= 2xy^2\vec{i}+ 2x^2y\vec{j}$ is a vector pointing in the direction of fastest increase of f with length equal to the rate of change of f in that direction. We can think of " $dx\vec{i}+ dy\vec{j}$" as a vector a "small" distance, dx, parallel to the x-direction, and a "small distance", dy, parallel to the y-axis. So $df= 2xy^2dx+ 2x^2ydy$ is the dot product of those two vectors, the rate of change of f along this "infinitesmal" vector.
• April 12th 2013, 06:32 PM
rickrishav
Re: Physical Significance of d(f(x,y))
Thanks a lot for your reply.
Just extending your view, and putting in my thoughts, we can think http://latex.codecogs.com/png.latex?...=%20x^2y^2%20a as http://latex.codecogs.com/gif.latex?f(x,y)=z, where z is the third axiz in the 3D- Plane. So the differential equation corresponds to the tangential plane at any point http://latex.codecogs.com/gif.latex?(x,y). Then, it can be seen as "the rate of change of f along this "infinitesimal" vector", just as we could map a tangent in a 2-D curve with rate of change of y with respect to x.

Thanks a lot @HallsofIvy. Your reply was extremely helpful.