12 votes

Interpreting a notation in calculus of variations (differentiating with respect to a derivative)

Consider a functional $J[y]$ defined by: $$J[y] = \int_a^b F(x, y, y') dx \tag{1}$$ I think it will be helpful to remind you right away that your notation implies that you have already agreed to ...
hft's user avatar
  • 556
5 votes

Proving function is differentiable at $(0,0)$ using total derivative definition

You can use that $\|h\|_2 = \sqrt{h_1^2+h_2^2}$ for $h\in\mathbb{R}^2$. If we now have a function $f:\mathbb{R}^2\longrightarrow\mathbb{R}$ with $f(0)=0$ and we want to prove that $f'(0)=0$, this ...
Nuke_Gunray's user avatar
  • 2,831
5 votes
Accepted

Iterated Integral of $(8x^3-36x^2y^2)dydx$

The order of integration is from the inside out, so $dy \, dx$ means you integrate with respect to $y$ first, then with respect to $x$. So the first antiderivative is $$\int 8x^3 - 36 x^2 y^2 \, dy = ...
heropup's user avatar
  • 136k
5 votes

Source of the definition of integrating a form along a curve in a manifold

This notion is known as a "line integral" and you should have already encountered it in the special case that $M=\mathbb{R}^n$ in a lecture on analysis. Lee has an entire chapter on this ...
Thorgott's user avatar
  • 11.7k
5 votes

Multivariate chain rule (again) for differential forms

Somehow I have figured it out only 15 minutes after asking the question, despite spending an hour on it before asking. The main observation is that $$ \sum_i \int \frac{\partial f_i}{\partial x_j}(t\...
Gareth Ma's user avatar
  • 3,737
4 votes
Accepted

Nonlinear homogeneous map $f : \Bbb{C} \times \Bbb{C} \to \Bbb{C}$

Consider an arbitrary continuous function $g$ on $\mathbb C$ st $g(0)=0, g(z)=z, |z| \ge 1, |g(z)| \le |z|$ but $g$ is not the identity. (eg take $g(re^{it})=r^2e^{it}, r \le 1, g(z)=z, |z| \ge 1$) ...
Conrad's user avatar
  • 27.5k
4 votes

Why $W$ is open in baby Rudin 2.28

Take $\mathbf y\in W$. Then, $(\mathbf 0, \mathbf y)\in V$. Since $V$ is open, find $\varepsilon>0$ such that $$\mathrm{B}_\varepsilon(\mathbf0,\mathbf y)\subset V$$ Then, you claim $\mathrm{B}_\...
ultralegend5385's user avatar
3 votes

Generalised integral $\int_{0}^{\infty} \int_{0}^{y} \sqrt{x^2 + y^2} e^{-x^2 - y^2} dx dy$

If we perform the change of integration variables from $x$ to $t$ via $x=yt$, we obtain $$ \int_0^{ + \infty } { \int_0^1 {y^2\sqrt {1 + t^2 } {\rm e}^{ - y^2(1 + t^2 )} {\rm d}t} \,{\rm d}y} . $$ ...
Gary's user avatar
  • 32.1k
3 votes
Accepted

The set of irrationals in $[0,1] \times [0,1]$ has measure zero in $\Bbb{R}^2$?

Presuming we talk about the Lebesgue-measure: Note that $\mu([0,1] \cap \mathbb{Q})=0$, so using $\mathbb{R}=\mathbb{Q} \cup \mathbb{I}$. $$1= \mu([0,1])= \mu([0,1] \cap \mathbb{R} )= \mu([0,1] \cap \...
tychonovs-scholar's user avatar
3 votes

Figuring out if $\lim_{(x,y)\to(0,0)}\frac{-x^6y^1(x^2+1)}{(x^6+y^2)\sqrt{x^2+y^2}}$ exists

Since this term has some $(x^2+y^2)$ term, it may be inspiring to use polar coordinate to investigate its behaviour first. Thus, under polar coordinate, we have the limit becomes $$\lim_{r\to0}\dfrac{...
Angae MT's user avatar
  • 1,031
3 votes
Accepted

Understanding $\frac{\text{d}u}{\text{d}t}$ vs. $\frac{\text{D}u}{\text{D}t}$

Pedantic, I know, but fluids are my area of expertise so I really can't let this go. $\frac{\mathrm d}{\mathrm dt}$ and $\frac{\mathrm D}{\mathrm Dt}$ are not equivalent operators. The operator $\frac{...
K.defaoite's user avatar
  • 12.5k
3 votes

When does a smooth vector field induce an ODE?

Too long for a comment. A vector field $v$ always induces an autonomous first order ODE in two dimensions $$\tag{1} \pmatrix{\dot x\\\dot y}=v(x,y)\,. $$ I am not sure why that seems not what you want ...
Kurt G.'s user avatar
  • 14.2k
3 votes
Accepted

Please explain Figure 24.4 in "Analysis on Manifolds" by James R. Munkres.

Yes, the blue surface is $A \cap M$ and the solid enclosed by the red closed curve $A \cap N$. The image of $F$ doesn't have to be rectangular, but you can always achieve this by restricting the ...
psl2Z's user avatar
  • 2,558
3 votes
Accepted

Showing that $ f(x) = 0 $ if the triple integral $\frac{f(x)}{(1+x^2+y^2+z^2)} $ converges

I'm not sure that will work since the range of the $a$ and $b$ variables will shrink as you go outwards, so that the integrals in those variables might go to zero. It might be better to use that ${\...
Zarrax's user avatar
  • 45.1k
3 votes

Deal with discontinuity in double integrals

$f$ is obviously not well-defined at $(0,0)$, and we can show that $$\lim_{(x,y) \to (0,0)} f(x,y)$$ does not exist for any path in $D$, since if $y = kx$ for some $k \in [0,1]$, we have for $x > 0$...
heropup's user avatar
  • 136k
3 votes

Interpreting a notation in calculus of variations (differentiating with respect to a derivative)

The "functional" is a way to convert functions to real numbers : It maps a function (of 1 variable or 2 variables or more variables) to a real number. Here the functional is $J[y]$ to ...
Prem's user avatar
  • 9,809
2 votes
Accepted

Optimization of 3 variable function when all variables have value

If we take the partial derivatives in each direction, we get the following: $$\begin{eqnarray} \frac{\partial f}{\partial x} & = & \frac{y(1 - z)}{\left(xy + (1 - z)(1 - x)\right)^2} \\ \frac{\...
ConMan's user avatar
  • 24.3k
2 votes

Help computing $\nabla_{X} \mathsf{tr}\left( f \left( X \right) Y \right)$

Trace is linear, so calculating any directional derivative, $\nabla_v \text{tr}(f(X)Y) = \text{tr}\big((\nabla_v f(X))Y\big)$.
Ted Shifrin's user avatar
2 votes
Accepted

If rank($J_f(x))=n$, then there exists a neighborhood such that for each of its points $y$, $rank(J_f(y))\geq n$

Hint: the function $$A\longmapsto\hbox{square submatrix of }A\longmapsto\det(\hbox{square submatrix of }A)$$ is continuous.
Martín-Blas Pérez Pinilla's user avatar
2 votes

What is the significance of defining the partial derivative as a one-sided limit or a two-sided limit?

You can consider the function $f(x)=|x|$ (of a single variable) and notice how the two definitions give different answers. Adding additional variables doesn't change anything.
quarague's user avatar
  • 5,921
2 votes

In geometric algebra, what is the dot product of a vector and a scalar? what is the wedge product of a vector and a scalar?

The issue is that the "dot product" isn't theoretically nice. There are two definitions I've seen. In terms of the geometric product, the first is $$ A_r\cdot B_s = \langle A_rB_s\rangle_{|...
Nicholas Todoroff's user avatar
2 votes

In geometric algebra, what is the dot product of a vector and a scalar? what is the wedge product of a vector and a scalar?

The general dot product formula for two k-vectors $ a_r, b_s $, of grades r and s respectively, is typically defined as a grade selection of the following sort: $$a_r \cdot b_s={\left\langle{{ a_r b_s ...
Peeter Joot's user avatar
  • 2,746
2 votes

Multivariable limit $\lim_{{(x,y) \to (0,0)}} \frac{x^2y^2}{x^2+y^4}$

Notice that $$\frac{x^2y^2}{x^2+y^4} \leqslant \frac{x^2y^2}{x^2}=y^2$$ and hence the function is bounded near $(0,0)$ by $y^2$. So, the limit is $0$.
Sayan Dutta's user avatar
  • 8,841
2 votes

Is the function $(x^2+y^2)\sin{(\frac{1}{x^2+y^2})}$ differentiable at the point $(0,0)$?

Since you already showed that $\frac{\partial{f}}{\partial{x}}$ and $\frac{\partial{f}}{\partial{y}}$ both exist at the point $(0,0)$ and are equal to $0$, your theorem 1 indicates that if $f$ is ...
Anne Bauval's user avatar
  • 35.3k
2 votes
Accepted

Semantics of the angle between velocity vector and the positive $x$-axis

You almost said it yourself: "if the angle is constrained to be in the range...". Therefore, by contrapositive (or contradiction), it must be that the angle is not constrained to any bounded ...
Mark S.'s user avatar
  • 24k
2 votes
Accepted

Classifying critical points when hessian has det0

$(0,0)$ is a stationary point for which the Hessian is $\begin{pmatrix} 2 & 0 \\ 0 & 0 \end{pmatrix}$, with eigenvalues $2$ and $0$. Now note that $$f(x,0) = x^2(1-x^2).$$ So increasing $x$ ...
kipf's user avatar
  • 2,357
2 votes
Accepted

Total derivative of f(x, g(x, y)) and its approximation

Aside: Higher Order Terms Why do I not need to consider higher order terms? Looking at Taylor Series would it make it more accurate? This is a largely-unrelated question that I will not address here;...
Mark S.'s user avatar
  • 24k
2 votes
Accepted

Finding the tangent line to curve to the ellipse $(x-3)^2+\frac{(y-4)^2}{4}=1$ through the origin

From your first method, you know that a vector of the form $$\vec t = \left(-\frac{1}{2} k(b-4), 2k(a-3) \right)$$ is tangent to the ellipse at the point $(a,b)$. Such a tangent will pass through the ...
heropup's user avatar
  • 136k
2 votes
Accepted

What does a one form $d\theta$ on $S^1$ do to $v \in T_p(S^1)$?

If $p\neq-1$ and $\phi:(-1,1)\to S^1\setminus\{-1\},t\mapsto\exp(\pi i t)$ then the derivation $\alpha\in T_pS^1$ is mapped by $d\theta$ to $\pi \cdot dx(\alpha\circ\phi^{-1})$. Just because $\phi^\...
FShrike's user avatar
  • 40.4k

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