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pasmith's latest activity
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pasmith
replied to the thread
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Riemann integrability and uniform convergence
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We have for n > N that (dropping the constant multiple of \epsilon) |f_n(x) - f(x)| < \epsilon. We can write this in the following two...
Today, 9:33 AM
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pasmith
replied to the thread
I
3-parameter group of symmetries on the plane
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E(2) also includes reflections. The rotations and translations generate a subgroup SE(2) of this group. E(2) is a semidirect product of...
Today, 6:09 AM
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pasmith
replied to the thread
Find the field inside and outside a spherical geometry
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Note that \mathbf{J} is zero everywhere except on r = a where it is not defined, but causes a jump in \mathbf{B} across r = a. Therefore...
Yesterday, 1:08 PM
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pasmith
replied to the thread
Continuous functions on metric spaces part 2
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In fact we have uniform continuity, since \delta = \epsilon works independently of x.
Yesterday, 11:05 AM
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pasmith
replied to the thread
Continuous functions on metric spaces part 2
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We are not even given that X is a vector space, so we don't have a concept of linearity. @Lambda96: All you have available to you are...
Saturday, 1:04 PM
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pasmith
replied to the thread
Continuous functions on metric spaces
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You should start a new thread. Note that this new question is a generalization of the question you posted in this thread.
Saturday, 9:38 AM
P
pasmith
replied to the thread
Continuous functions on metric spaces
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|\phi(f) - \phi(g)| = \left| \int_a^b f(x)\,dx - \int_a^b g(x)\,dx \right| = \left| \int_a^b f(x) - g(x)\,dx\right| etc.
Friday, 2:04 PM
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pasmith
replied to the thread
A
Heat conduction equation in cylindrical coordinates
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Define "hollow". Do you mean a circular annulus lying between 0 < a \leq r \leq b? In that case yes, both J and Y are admissible...
Friday, 12:23 PM
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pasmith
replied to the thread
Continuous functions on metric spaces
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Hint: By basic properties of the Riemann integral, \left| \int_a^b f(x)\,dx \right| \leq \int_a^b |f(x)|\,dx \leq (b- a) \sup |f|.
Friday, 9:54 AM
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pasmith
replied to the thread
Fundamental matrix for complex linear DE system
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You know from the Cayley-Hamilton theorem that A = \begin{pmatrix}2 & - 5 \\ 4 & - 2 \end{pmatrix} satisfies its own characteristic...
Friday, 4:48 AM
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pasmith
replied to the thread
Find the values of a and b to get max/min of certain expression
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Note that sapping a and b swaps 3a^2 + 2b and 3b^2 + 2a, leaving the maximum invariant. Note also that 3a^2 + 2b is strictly increasing...
Thursday, 11:30 AM
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pasmith
replied to the thread
Solving non-homogeneous system of ODE using matrix exponential
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No. The indefinite integral already includes an arbitrary constant; you don't need to add an e^{At}x(0) term in this case. That leaves...
Wednesday, 3:09 PM
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pasmith
replied to the thread
Solving non-homogeneous system of ODE using matrix exponential
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They are evidently using e^{At}\int^t e^{-As}f(s)\,ds with an indefinite integral, so that an arbitrary constant of integration must be...
Wednesday, 10:04 AM
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pasmith
replied to the thread
Prove the given hyperbolic trigonometry equation
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You need to end up with \sinh^2 \frac x2, so start from \cosh 2x - 1 = 2\sinh^2 x. But using these identities comes very close to...
Wednesday, 8:21 AM
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pasmith
replied to the thread
True or false problem for double differentiable function
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It i a counter example to the assertion f''(x) > 0 \Rightarrow f(x) > 0.
May 13, 2024
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