Differentiability of a function of two variables

  • #1
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Main Question or Discussion Point

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?
 

Answers and Replies

  • #2
Math_QED
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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)
 
  • #3
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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?
 
  • #4
hilbert2
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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.
 
  • #5
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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 possibilty 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.
 
  • #6
hilbert2
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Yes, I should have emphasized that the point ##x## where the derivative is evaluated is kept constant and only the ##\Delta x## is varied.
 
  • #7
Stephen Tashi
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,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.
 
  • #8
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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.
 
  • #9
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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.
 

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