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Hi all,

In my Calculus III course, we are using Stewart's book, so as you might know there is not much rigor in there.

Likewise, when it came to the section on Maximum and Minimum values of a function with two variables z=f(x,y), they ommited a lot of stuff.

Hence, i tried to use simmilar ideas as in single variable calculus, to derive some of the theorems and also tried to prove them as well concerning some topics.

Specifically, i started out by first trying to define what it means for a function of two variables to be concace up or down, then moved on trying to establish a theorem(equivalent to single-variable calc) that says if the first derivative is >0,<0 then the function is increasinng/decreasing respectively. But in this case, i used the idea of directional derivatives.

And then moved on to constructing another theorem that says if the second directional derivative of f is >0 then the function is C.U.(again here, extrapolating from single-variable calculus).

The statements of these theorems and their proofs are attached.

Bear in mind, i have never seen anywhere neither these theorems nor their proofs, i just tired to extrapolate from single-variable calculus, so my question is do they make any sense? I mean are they mathematically correct?

All the best!

In my Calculus III course, we are using Stewart's book, so as you might know there is not much rigor in there.

Likewise, when it came to the section on Maximum and Minimum values of a function with two variables z=f(x,y), they ommited a lot of stuff.

Hence, i tried to use simmilar ideas as in single variable calculus, to derive some of the theorems and also tried to prove them as well concerning some topics.

Specifically, i started out by first trying to define what it means for a function of two variables to be concace up or down, then moved on trying to establish a theorem(equivalent to single-variable calc) that says if the first derivative is >0,<0 then the function is increasinng/decreasing respectively. But in this case, i used the idea of directional derivatives.

And then moved on to constructing another theorem that says if the second directional derivative of f is >0 then the function is C.U.(again here, extrapolating from single-variable calculus).

The statements of these theorems and their proofs are attached.

Bear in mind, i have never seen anywhere neither these theorems nor their proofs, i just tired to extrapolate from single-variable calculus, so my question is do they make any sense? I mean are they mathematically correct?

All the best!

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