Understanding Derivative and Integral Notation in Partial Differentiation

[clm222]

I just started partial differentiation, and (amoung a few others) it brings up some questions of notation.

my first few questions are for the integral.

first: doesn't the "dx" at the end of an integral mean "in respect to x"? or any toher variable like "dl", in respect to l?

ie: $\int 4x-xj dx=2{x^2}-\frac{j{x^2}}{2}$?
$\int 4x-xjdj=-x$

second: if i want to do definite integration from a to b, and i have the second derivative, how to i diplay the lintegrand?

$\int_a^b \int f''(x)dx$?
or maybe $\int \int_a^b f''(x)$? I'm not sure

i also have some questions about derivatives, and their notaion.

first: is it bad to have a function 'd', since you will likely counter stuff like $\frac{dd}{dx}$?

second: for partial derivatives, is it still bad to use 'd', like in my last question?

third: what are the details of using Leibnez's notation for higher order derivatives. can I write (given f(x,y)=z) "$f_{xx}$" as "$\frac{∂f}{∂x∂x}$"? or as"$\frac{∂f}{∂{x^2}}$". same with, say: $f_{xyy}=\frac{∂f}{∂x∂{y^2}}$ or $f_{xxyxx}=\frac{∂f}{∂{x^2}∂y∂{x^2}}$
same with the "$f_x$" notation. does $f_{xx}=f_{x^2}$? $f_{yxx}=f_{y{x^2}}$ or $f_{xxyxx}=f_{{x^2}y{x^2}}$?

please correct me any of my mistakes, i am not fully familiar with these notations. thank you.

 

[Bacle2]

There's some info here:

http://en.wikipedia.org/wiki/Partial_derivative

 

[HallsofIvy]

I just started partial differentiation, and (amoung a few others) it brings up some questions of notation.

my first few questions are for the integral.

first: doesn't the "dx" at the end of an integral mean "in respect to x"? or any toher variable like "dl", in respect to l?

ie: $\int 4x-xj dx=2{x^2}-\frac{j{x^2}}{2}$?
$\int 4x-xjdj=-x$

Was this a misprint? $\int 4x- xj dj= 4j- xj^2/2$ (plus a constant of course). It looks like you accidently differentiated rather than integrated.

second: if i want to do definite integration from a to b, and i have the second derivative, how to i diplay the lintegrand?

$\int_a^b \int f''(x)dx$?
or maybe $\int \int_a^b f''(x)$? I'm not sure

I'm not clear what you are talking about. Why would you have limits on one integral and not the other? What are you trying to find here? In any case, to have a double integral you really need to have two different variables.

i also have some questions about derivatives, and their notaion.

first: is it bad to have a function 'd', since you will likely counter stuff like $\frac{dd}{dx}$?

Yes, avoid using 'd' for anything other that the differential sign! If your function is a "distance", use "D".

second: for partial derivatives, is it still bad to use 'd', like in my last question?

The danger of misunderstanding is not as bad but it would still be better to use the "correct" notation, $\partial$ (the Latex code is "\partial")

third: what are the details of using Leibnez's notation for higher order derivatives. can I write (given f(x,y)=z) "$f_{xx}$" as "$\frac{∂f}{∂x∂x}$"? or as"$\frac{∂f}{∂{x^2}}$". same with, say: $f_{xyy}=\frac{∂f}{∂x∂{y^2}}$ or $f_{xxyxx}=\frac{∂f}{∂{x^2}∂y∂{x^2}}$
same with the "$f_x$" notation. does $f_{xx}=f_{x^2}$? $f_{yxx}=f_{y{x^2}}$ or $f_{xxyxx}=f_{{x^2}y{x^2}}$?

Yes. And, as long as the derivatives are continuous, the order of differentiation is not important.

please correct me any of my mistakes, i am not fully familiar with these notations. thank you.

 

[clm222]

Was this a misprint? $\int 4x- xj dj= 4j- xj^2/2$ (plus a constant of course). It looks like you accidently differentiated rather than integrated.


I'm not clear what you are talking about. Why would you have limits on one integral and not the other? What are you trying to find here? In any case, to have a double integral you really need to have two different variables.


Yes, avoid using 'd' for anything other that the differential sign! If your function is a "distance", use "D".


The danger of misunderstanding is not as bad but it would still be better to use the "correct" notation, $\partial$ (the Latex code is "\partial")


Yes. And, as long as the derivatives are continuous, the order of differentiation is not important.

thats embarassing, i did differentiate, my bad :S

 

[clm222]

ok thank you

i acually just learned last night what a integral really was, i thought integration was antidifferentiating with the limits, i didnt know that integration is finding the difference of the antiderivatives to calculate the areas, not the difference of the function its self.