My original equation was taken from something with messy notation, and I lost some subscripts while working part of the larger question out. So I gave you a poorly framed question to answer -- I'm sorry.
In my original question, the function f(x,y) is vector valued, and the integrand is...
I don't know how to apply Leibniz's rule for this integrand. I believe there is a substitution to allow me to express this integral differently, but the partial being a function of what we're integrating by is confusing me.
\int^{b}_{a} \frac{\partial f(x,y)}{\partial y(x)} dx
Knowns are that...
So as an example, if defining g(x)=f(x^{2}+45x-2) and Eisenstein's criterion showed that g(x) is irreducible, then f(x) is irreducible? Or can I only use linear factors such as g(x)=f(x-2)?
This is for clarification of a method.
Dummit & Foote, pg 310, Example (3).
f(x)=x^{4}+1 is converted into g(x)=f(x+1) in order to use Einsenstein's Criterion for irreducibility. The example states "It follows that f(x) must also be irreducible, since any factorization of f(x) would...
So there's no hangup working on the whole real line from the start due to the derivative being bounded? I was unsure about using the MVT in a more general setting.
Homework Statement
Prove f(x)=\sqrt{x^{2}+1} is uniformly continuous on the real line.
Homework Equations
Lipschitz Condition: If there is a constant M such that |f(p) - f(q)| \leq M |p-q| for all p,q \in D, then f obeys the Lipschitz condition.
Mean Value Theorem: Let f be continuous on...
I found the fundamental property regarding splitting up piecewise smooth curves two chapters earlier, which I need to review. I got an answer of -\pi, which is what the solutions suggested. Thank you very much!
Can this be done finitely many times as long as the pieces form a closed curve when put together? Are there other stipulations for breaking apart the curve to work, or onlyclosure of the curve?
Analyzing an integral over a non-exact region for gamma defined by |x|+|y|=4
The following was similar to a problem on a calculus final that I got wrong. It is an extension of a problem in R.C. Buck "Advanced Calculus" on page 501. Similar to knowing the trick to integrating e^{|x|} (which...
The definition I have for a random variable is
X=\lbrace \omega \in \Omega \vert X(\omega) \in B \rbrace \in F where F is a sigma algebra and B is a Borel subset of R.
Using function composition, how would one write a similar set notation definition for f(X), where f is a Borel measurable...
HallsofIvy,
I'm trying to show that for F_{n} \subset F_{n+1} for all n (they're sigma-algebras), that \cap ^{\infty}_{i=1}F_{i} may not be a sigma algebra. Counter example is what the exercise says.
I was thinking of a sequence of intervals that never include 0. But I didn't know how to...
I guess I've fallen through some of the cracks in the plethora of definitions I've learned, or I just never had enough examples of taking limits of intervals. Anyways, which is true, and why?
$\cap^{\infty}_{n=1}(0,\frac{1}{2^{n-1}}]=0?$...
I can't exactly "fix my constraints". I'm trying to prove something as generally as possible. Can you think of any constraints I can add that would easily establish the desired result? Because I can't.