Differentiability of a function of two variables

[Jazzyrohan]

I have been studying multivariable calculus but I can't quite think visually how a function will be differentiable at a point.

How can a function be differentiable if its partial derivatives are not continuous?

 

[member 587159]

This can even fail in the 1 dimensional case.

Consider $f(t) = t^2 \sin(1/t)$ when $t \neq 0$ and $f(0) =0$. This function is differentiable on $\mathbb{R}$ but its derivative isn't continuous.

Why would the result hold in the higher derivative case? (Similar counterexamples are possible)

 

[Jazzyrohan]

Got it but directing to my first doubt how can differentiability be defined for a function of two variables,like,what is the basic condition for such a function to be differentiable?

 

[hilbert2]

For a real-valued function of one real variable, the differentiability means that the difference $f(x+\Delta x) - f(x)$ can be arbitrarily well approximated by a differential $f'(x)\Delta x$, and the error of this approximation decreases as fast as $(\Delta x)^2$ when the $\Delta x$ is approaching zero. The real number $f'(x)$ is the derivative.

For a real-valued function of two variables, the equivalent definition is that the difference $f(\mathbf{x}+\mathbf{\Delta x}) - f(\mathbf{x})$ can be similarly approximated with a dot product $\mathbf{f'(x)}\cdot\mathbf{\Delta x}$. Here the derivative $\mathbf{f'(x)}$ is now a two-component vector and it is the same as the gradient of the function $f(\mathbf{x})$.

If the function takes two arguments and returns a two-component vector, the derivative $\mathbf{f'(x)}$ is a $2\times 2$ matrix and the differential is $\mathbf{f'(x)}\mathbf{\Delta x}$, where there's a matrix-vector multiplication instead of a dot product.

 

[fresh_42]

Got it but directing to my first doubt how can differentiability be defined for a function of two variables,like,what is the basic condition for such a function to be differentiable?

Maybe the beginning of https://www.physicsforums.com/insights/the-pantheon-of-derivatives-i/ can help you here. As @hilbert2 has said: differentiability is only the possibility to approximate a function locally by a linear function. The question, whether these many local events result in a continuous dependency of the location is a completely different one. The most common mistake comes from the notation: $f\,'(x)$. It is wrong. It should better be $f\,'(a)$, since this is what differentiation does: it gives a slope at a certain point. Differentiability now means, that this can be done at all points $x=a$, such that we get a new function $a \longmapsto f\,'(a)$ which people write $x \longmapsto f\,'(x)$ and since people are lazy, abbreviate it by $f\,'(x)$. The result is, that the dependency of the location, at which $x \longmapsto f(x)$ has been approximated by a linear function via the calculation $\left.\dfrac{d}{dx}\right|_{x=a}f(x)$ is completely lost.

So what differentiablity means is that those linear approximations exist, at a point or everywhere.
Now whether this differentiabilty depends continuously or differentiable from said location is another step.

 

[hilbert2]

Yes, I should have emphasized that the point $x$ where the derivative is evaluated is kept constant and only the $\Delta x$ is varied.

 

[Stephen Tashi]

,what is the basic condition for such a function to be differentiable?

Are you asking for necessary and sufficient conditions for a function of several variables to be differentiable? As other's have indicated, those conditions are that it satisfies the definition of being differentiable. That's a trivial answer, but I don't know a way to express such conditions in terms of simpler concepts.

 

[fresh_42]

Yes, I should have emphasized that the point $x$ where the derivative is evaluated is kept constant and only the $\Delta x$ is varied.

I didn't want to criticize you. My post has been meant as a general reminder, because there are so many different ways to regard a differentiation, and that short notations often disguise the individual view. I once listed a few views of it and found $10$: first page here https://www.physicsforums.com/insights/journey-manifold-su2mathbbc-part/ and the word "solpe" didn't even occur in the list. It was fun to see how a simple tangent can have so many roles.

 

[Jazzyrohan]

I was confused about a few things and I think I get it now.Heartiest thanks to all of you .I really do appreciate your help.
Can you also tell me some books containing good theory or visualization and a few books for problem solving?I am currently in first year of college.