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Jul
19
comment Let $A,B$, and $C$ be sets. If $A\subseteq B$, $B\subseteq C$, and $C\subseteq A$, then $A=C$.
I'm trying to hint that you may already know a little proposition about arbitrary sets $X$ and $Y$ that reduces the proof of what you want to just invoking that proposition. What propositions do you already know about the subset relation?
Jul
19
comment Compute Surface Integral
As @Ivo Terek indicates in his (accepted) answer, when you integrate a scalar field over a surface, you do not take its dot product with the normal—as indeed you note would make no sense.
Jul
19
comment Tangent plane and tangent lines to curves through a point
Can you offer of such an example where there are no such smooth curves??
Jul
18
comment Show the function is continuous in $\Bbb R^2$
To compute the partial derivatives at $(0,0)$, you'll need to go back to their definition in terms of limits—and this would seem to be no simpler than dealing showing $\lim_{(x,y) \to (0,0) f(x, y) = 1$.
Jul
18
comment Compute Surface Integral
If the question is to integrate $x^2 + y^2$ then you are not integrating a vector field but rather the scalar field $F(x, y, z) = x^2 + y^2$.
Jul
18
comment Let $A,B$, and $C$ be sets. If $A\subseteq B$, $B\subseteq C$, and $C\subseteq A$, then $A=C$.
Do you already have at your disposal the little result that if $X \subset Y$ and $Y \subset X$, then $X = Y$?
Jul
18
comment If we have an embedding $f:X \rightarrow A$, where $A \subset Y$, do we have to show $f^{-1}$ is continuous?
"Embedding" may be used in more restrictive ways in a particular context, e.g., an isometric embedding. But if all you have are topological spaces, or all you're considering is topology, then "embedding" is synonymous with "topological embedding".
Jul
18
comment Does this intuition for “calculus-ish” continuity generalize to topological continuity?
As to your students' question as to why the projections should be continuous: Is that not a natural generalization of the situation the projections from the cartesian plane onto each of the coordinate axes?
Jul
18
comment Does this intuition for “calculus-ish” continuity generalize to topological continuity?
The issue for a function $f: X \to Y$ between topological spaces is: what is the meaning of $\lim_{x \to c} f(x)$? One may use the notion of "net" to formulate the definition and then, with that definition, the condition you state is equivalent to continuity at $c$, provided you state it a bit more precisely: for every net $(x_i)_{ \in I}$ that converges to $c$ in $X$, the net $\bigl((f(x_i)\bigr)_{i \in I}$ converges to $f(c)$ in $Y$.
Jul
12
revised Tangent plane and tangent lines to curves through a point
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Jul
12
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Jul
12
revised Tangent plane and tangent lines to curves through a point
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Jul
11
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Jul
11
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Jul
11
asked Tangent plane and tangent lines to curves through a point
Jul
6
comment How can $f(x)=x^4$ have a global minimum at $x=0$ but $f''(0)=0$?
@Whyka: right, $a > 0$ implies $a^b > 0$ for all real $b$. In your example, $f(x) > 0$ for all $x \neq 0$ and $f{0} = 0$, which is why I gave the condition $b > 0$.
Jul
6
comment How can $f(x)=x^4$ have a global minimum at $x=0$ but $f''(0)=0$?
@Whyka: Certainly if $a > 0$ and $b > 0$, then $a^b > 0$.
Jul
5
comment Solve 4 A (L^(3/4)) - wL (((24 - L) w)^(-3/4)) = -(((24 - L) w)^(1/4)) for L? Using Mathematica
The code has also a second error: wL should be w L.
Jul
5
comment How can $f(x)=x^4$ have a global minimum at $x=0$ but $f''(0)=0$?
"maximum" should be "minimum" in the answer at hand. How show this function has a minimum at $x=0$? simple: use the definition of "minimum"!
May
1
comment Traditional axes in 3d Mathematica plots?
"Traditional axes" to me sounds like the way mathematicians would typically sketch a 3D plot, no just with axes emanating from the origin, but with arrows at their positive ends and axis labels, too.