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

What is the physical meaning of *d(x*^{2}y^{2})?

For example,

*d(x*^{2}) = 2x dx

This can be derived as,

d(x^{2})/dx = 2x

d(x^{2}) = 2x dx

Here *dx* can be thought of *Delta x*, rather than thinking *d/dx* as an operation. So, small change in *x*^{2} is the **product of "two times current value of x" and small change in ****x**.

What will be the similar deduction methodology for* d(x*^{2}y^{2}) = 2xy^{2}dx + 2yx^{2}dy?

Thanks a lot.

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

Given $\displaystyle f(x,y)= x^2y^2$ then $\displaystyle grad f= \nabla f= 2xy^2\vec{i}+ 2x^2y\vec{j}$ 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 "$\displaystyle dx\vec{i}+ dy\vec{j}$" as a vector a "small" distance, dx, parallel to the x-direction, and a "small distance", dy, parallel to the y-axis. So $\displaystyle df= 2xy^2dx+ 2x^2ydy$ is the dot product of those two vectors, the rate of change of f along this "infinitesmal" vector.

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.