Santiago Canez
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 Mar31 comment Proving that $P[0, 1]$, the space of all polynomials on $[0, 1]$ is not complete. Note that you don't necessarily have to show that the given sequence converges to $e^x$, just that it doesn't converge to a polynomial, which might be simpler depending on what you know about Taylor series. Mar29 comment Find all points on a surface which have a tangent plane parallel to given plane - is my method correct? Also, when coming up with $y=x-\frac23$ it seems you divided by $x+y$ at some point, meaning that doing this ignores possible points for which $x+y=0$. So, you should consider these points separately at the end, meaning use $y=-x$ and $f_x=6$ to find precise points. Mar29 comment Find all points on a surface which have a tangent plane parallel to given plane - is my method correct? Using only $f_x=-f_y$ isn't enough since you're ignoring that you have actual values for each of these, 6 for the first and -6 for the second. (Again, $k$ must be $1$ as you can determine by looking at the $z$-components of the gradients.) Your solution gives points for which $f_x=-f_y$ but which don't satisfy $f_x = 6$ and $f_y = -6$ as required. Once you have $y=x-\frac23$ you can then use $f_x=6$ for solve for $x$ to narrow down your values. Mar29 comment Find all points on a surface which have a tangent plane parallel to given plane - is my method correct? @HelenByrne, see my comment to your original question. Mar29 comment Find all points on a surface which have a tangent plane parallel to given plane - is my method correct? It doesn't make sense for the normal vectors to be vectors in $\mathbb R^2$ as you assume when you use $f_xe_1+f_ye_2$. You can't forget about the $z$-component, meaning that you should take the gradient of the three-variable function $f(x,y,z)=x^2-y^3-2xy-z$ of which your surface is a level surface, and similarly for the given plane. (In other words, you can't move the $z$ term to the right as you did.) The resulting gradients should be parallel yes, but by comparing the $z$-coordinates you can conclude that they must be equal, as the solution you commented on below suggests. Mar28 comment If $f(x)=\tilde{f}(\|x\|)$ and $f$ is continuous, is $\tilde{f}$ continuous? @quid, fair enough. Mar28 comment If $f(x)=\tilde{f}(\|x\|)$ and $f$ is continuous, is $\tilde{f}$ continuous? So $\tilde f$ is defined on $[0,\infty)$? You can view it as the restriction of $f$ to the nonnegative $x_1$-axis, right? Mar27 comment Finding the Matrix of a Given Linear Transformation T with respect to a basis Interesting. My guess at an interpretation was that the OP is looking at the linear transformation from the space of $2 \times 2$ upper-triangular matrices to itself given by multiplication by $A$, but perhaps your interpretation is correct. Mar27 comment Finding the Matrix of a Given Linear Transformation T with respect to a basis What is $T$? What is its domain and codomain? What space is the basis you provided a basis of? Mar27 comment Prove $\int_{a}^{b}{f(x)dx}>0$. Fair enough. Thanks. Mar27 comment Prove $\int_{a}^{b}{f(x)dx}>0$. How do you show this without (essentially) first proving the claim in the original post? Mar27 comment Prove $\int_{a}^{b}{f(x)dx}>0$. Saying that $\int_{x_0-\epsilon}^{x_0+\epsilon} f(x)\,dx > 0$ since $f(x) > 0$ for all $x \in (x_0-\epsilon,x_0+\epsilon)$ essentially relies on what the OP is asking to prove, no? Mar27 comment Minimal polynomial and diagonalization. Eigenvalues don't have to be distinct in order to be diagonalizable. Mar25 comment proving that the set of all english words is countble. A countable union of countable sets is also countable. Mar24 comment $E$ compact, real-valued $f : E \to \mathbb{R}$ continuous iff graph is compact - is real valued necessary? Saying that $g := (x,f(x))$ is continuous requires a metric on $E \times Y$ which is compatible with the product topology. So, yes, your proof does use it. Of course, there are different possible metrics with this property, but you should be clear that you are using such a metric. Mar24 comment $E$ compact, real-valued $f : E \to \mathbb{R}$ continuous iff graph is compact - is real valued necessary? No those assumptions aren't necessary, but be careful with your wording. Saying that $x$ is a polynomial doesn't make sense on an arbitrary metric space, better to say that $x \mapsto x$ is the identity function on $E$. And second, for $g := (x,f(x))$ to be continuous the metric on $E \times Y$ is clearly relevant, so you should be clear about what metric you're using. Mar24 comment $E$ compact, real-valued $f : E \to \mathbb{R}$ continuous iff graph is compact - is real valued necessary? @ThomasE. A continuous bijection with compact domain and Hausdorff codomain has a continuous inverse, although this should be made clearer. Mar24 comment show $f_n(t)=t^n$ is cauchy What metric are you using? Mar24 comment Unit Vectors in Rotation Matrices What do you mean by "for my result to be correct"? Mar23 comment What is meant by $\frac{d ^2y}{dx^2}$? @GitGud, do you have a better notation for the "second-derivative operator" $\frac{d^2}{dx^2}$?