Recent content by sudhirking

This is an opinionated response. I would say Analysis re-examines those things conceived in calculus that had not the precise notion of "infinitesimals" which is ultimately found in limits. If we hear the subjects "calculus" we immediately think of Newton, Leibniz, and other pioneers and we...

To the original questions, are their really only 4 fundamental operations, there have been many responses. Some would say "4? Of course not at least 2. The inverse operations are just the normal operation on the inverse elements". And there were other comments on saying the counting addition is...

Regardless of the connections these ellipses have with parabolas, and how they are all conceivably unified under a similar theme, this question must be answerable as a negative. For we know the expression of a parabola, 0 = Ax2 + Bx1 + Cx0 - y. It is clearly a function in the sense that any...

I would like to answer a slightly different question. Suppose you had to integrate the first derivative of a function. How then would you proceed? Your response would be something along the lines of, "Well of course FTC! The integral of a derivative would be up to some constant, the...

Great inquiry and along your investigation you even noted the vulgarity of the notation! To free yourself from the corruption the present notation is proving, I recommend the following: you started from a function of x and y and then switched to new coordinates x' and y' by the...

I actually believe the OP made a slight typo again! I believe he wishes the transformation to be from 2-D to 3-D. Reason I infer this is from the following: * the initial post posited that the transformation was from 2-D to 2-D (I think in my first post you can see it in the quote). *The OP...

While the above technique is the most promising, as it makes reference to no other special theorem by itself, an alternative popular approach is as follows: ask yourself what does this transformation do on the vector (1,0)? what does it do to the other basis vector (0,1)? These form...

I do not know what the term matrix means to you, but in brief and for our purposes, a matrix is simply a "linear transformation" which when acting on a vector gives out another vector! The size and shape of the matrix tells you the quality of vectors it takes in and those which it spits out...

Before answering, one rather urgent issue is raised: the linear transformation unaffects the condition of the input (e.g. takes a 2-D vector and gives out a 2-D vector) But then you define the transformation to spit out a vector that has 3-components? I must take that to mean that there is...

You ask a question which can be answered, but if we answer it for you it robs you the joy of discovering that yourself! So as others have said it is best recommended to practice and find when one technique is more suited than another. It helps not only to solve an ensemble of questions, but to...

Yes of course, and as the square of anything induces an inherent ignorance on the sign, the square root can make no inference on that subject. In fact the square had the property of taking any real number and putting out a real number. The mapping from one to the other via the square is not...

Hello and Welcome! I would like to motivate the nature of the logarithm so that you can understand why such a device is even created. Suppose I told you that a number square is 25. Is it not fair to ask the question what is that number (whose square is 25)? And should not, much like there...

The difficulty in the differentiation as you noted is there because of our ignorance on f(u)! But this problem is really no problem at all. For suppose, to bring light on the thing being missed, f(u) is something as simple as sin(u). Carry out derivatives if f(u) is beknown to the audience...

One way to go about the problem is to just compute the individual partial derivatives and exhibit their sum as zero. While this is brute force, when in doubt and when no simple principles seem to afford opportunity, the best you can do is just the obvious. After taking the derivatives it does...

Dear Johnqwertyful, Before we move on to the problem let us consider the following logic on natural numbers. It is agreed that the identity number 0 does nothing when added to some other number x. That is x +0 is of course x. And another way of writing 0 is by a number subtracted from its...