Looking for a Comprehensive Calculus-Diff.Eq Text: Any Recommendations?

[s0laris]

Hi guys,
I am wondering if anyone can recommend a good calculus-diff.eq text that does a good job at explaining concepts on an intuitive level. I have already covered the material, but many times I just learned to solve problems algorithmically, rather than understanding the theory. In turn, looking for a super thorough review. Any suggestions?

 

[paulfr]

The theory is not that hard.
Doing the problems, especially integration, is the hard part
Choosing which technique, which expression for U [in U Substitution] and
knowing what to do next in a long integration. That comes from doing problems
and seeing lots of solutions. And it demands a knowledge of previous Mathematics
which includes Trig Identities, Algebraic manipulation, Completing the Square,
Polynomial Long Division, Factoring, LCDs, etc.

But the big picture is quite simple.
Integration is just multiplication and the limit just makes the result of the summation [recall that multiplication is repeated summation] more accurate.

Similarly, Differentiation is just division in the limit and the limit also makes the result more accurate. And division is nothing more than repeated subtraction.

And the Fundamental Theorem tells us that the two operations are Inverse Functions with all that that implies.

Relax
You probably know more that you think you do.

 

[RobertT]

The theory is not that hard.
Doing the problems, especially integration, is the hard part
Choosing which technique, which expression for U [in U Substitution] and
knowing what to do next in a long integration. That comes from doing problems
and seeing lots of solutions. And it demands a knowledge of previous Mathematics
which includes Trig Identities, Algebraic manipulation, Completing the Square,
Polynomial Long Division, Factoring, LCDs, etc.

But the big picture is quite simple.
Integration is just multiplication and the limit just makes the result of the summation [recall that multiplication is repeated summation] more accurate.

Similarly, Differentiation is just division in the limit and the limit also makes the result more accurate. And division is nothing more than repeated subtraction.

And the Fundamental Theorem tells us that the two operations are Inverse Functions with all that that implies.

Relax
You probably know more that you think you do.

Don't be so sure. For people who study pure mathematics, they need to know everything down to the finest details to be 100% sure at what exactly they are doing even for a simple integration. Often, students do not even understand the underlying meaning of differentiation in terms of mathematics.

 

[s0laris]

Well, for example, I never had an intuitive understanding of a Lagrange multiplier or a Jacobian. I could only solve them through symbolic manipulation, which, of course, is no fun at all. And as RobertT said, I definitely need to know everything down to the finest details.