Questions tagged [multivariable-calculus]
Use this tag for questions about differential and integral calculus with more than one independent variable. Some related tags are (differential-geometry), (real-analysis), and (differential-equations).
35,897
questions
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16
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Local maxima of $f(x,y) = x^2 + y^2 - xy$ subject to $x^2 = 16$, $y^2 = 16$
Given the problem below:
Write down the Lagrangian when the problem is to maximise the function
$$
f(x, y)=x^2+y^2-x y
$$
on the set
$$
\{(x, y):-4 \leq x \leq 4,-4 \leq y \leq 4\} .
$$
Use the Kuhn-...
0
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0
answers
21
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Tangent Vector, Principal normal vector, Binormal vector, and Torsion
So I'm trying to fully grasp how all these relate. My current understanding is that the tangent vector describes the direction in which the curve is going/curving. Meanwhile, the principal norm is ...
1
vote
1
answer
16
views
Confusion about the units of divergence
For a 3-D vector field $V = (u \hat{\textbf{i}} + v\hat{\textbf{j}} + w\hat{\textbf{k}})$:
$$\nabla \cdot V\Big[\frac{m}{s}\Big] = \frac{\partial u [\frac{m}{s}]}{\partial x [m]} + \frac{\partial v[\...
- 467
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0
answers
60
views
Needing help re-deriving an equation in a paper
I am trying to understand where an equation came from in a paper.
Earlier, the paper gives these relations
$$\left(\nabla^2-\frac{1}{r^2} \right) u_r
+\frac{1}{1-2\nu} \frac{\partial e}{\partial r}
+ ...
- 11
1
vote
0
answers
37
views
Show equivalence of two very long integrals.
I'm trying to show that the following integral $$\int_0^\infty \int_{-\infty}^s f(x \vee s-b) \sqrt{\frac{2}{\pi}}\frac{1}{t^{\frac{3}{2}}}(2s-b)e^{-\frac{(2s-b)^2}{2t}-\mu(b+\frac{\mu t}{2})}db ds$$ ...
- 39
1
vote
0
answers
13
views
Logistic regression with gradient descent derivation
Our maximum likelihood is:
$$l(\beta)=\sum_{i=1}^{n}y\log(p(x))+(1-y)\log(1-p(x))$$
$$l(\beta) = \sum_{i=1}^{n} y^{(i)} \log(\sigma(\beta^\intercal x^{(i)})) + (1 - y^{(i)}) \log(1 - \sigma(\beta^\...
- 521
-1
votes
0
answers
28
views
Some Confusion with the Partial Derivative Chain Rule
In my calculus class, we were asked to show that in polar coordinates, Laplace's equation is: $$ \dfrac{\partial^2z}{\partial x^2} + \dfrac{\partial^2z}{\partial y^2} = 0 = \dfrac{\partial^2z}{\...
- 1
1
vote
1
answer
81
views
Finish this proof of gradient in polar coordinates
In my exercise, we want to derive the formula of the gradient in polar coordinates.
We end up showing that
$$\nabla f(r\cos \theta , r\sin\theta)=\frac{ \partial g }{ \partial r }(r,\theta)e_{r }+\...
- 169
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0
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13
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Differentiable function and zero derivative [closed]
I'm trying to solve the following question:
Consider a differentiable function $f: U \subset \mathbb{R}^{n} \longrightarrow \mathbb{R}$, where $U$ is open and connected. The differential $ df(p): \...
- 21
0
votes
5
answers
75
views
Help with integration over a triangular region.
I'm currently trying to wrap my head around double integrals over a triangular region, when you are given the vertices of the triangle. I need to do the integral
$$
\iint_D 2e^{-y-x} \,\mathrm{d}y\,\...
- 1
0
votes
0
answers
40
views
What integral is used to calculate the electric field generated by a continuous charged curve?
I'm studying Multivariable Mathematics, by Ted Shifrin, in which one reads that ''the gravitational force exerted by a continuous mass distribution $\Omega$ with density function $\delta$ is
$$\mathbf{...
0
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1
answer
56
views
A question about a multivariable limit
I have $$\lim_{(x,y)\to (1,0)} \frac{xy^2 -y^2}{(x-1)^2+3y^4}$$ where the answer must be "the limit does not exist".
I set $y=m(x-1)$ and get $$\lim_{x\to 1} \frac{m^2(x-1)}{1+3m^4(x-1)^2}=0....
- 2,807
0
votes
1
answer
29
views
How to write Jacobian matrix in more vector-like format
Let $\mathbf{f}: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ be a vector field. Suppose I wish to compute $\frac{\partial \mathbf{f(x)}}{\partial \mathbf{x}}$. Then we have the Jacobian.
$$\frac{\...
- 4,939
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0
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24
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How does the del operator work exactly? [duplicate]
In Cartesian coordinates, vector operations are as simple as if we were treating the del operator as like a vector.
$$\nabla = \Big(\frac{\partial}{\partial x}\hat{\textbf{i}} + \frac{\partial}{\...
- 467
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0
answers
18
views
How to solve simple second order ODE with RHS $x / \sigma^2$
I have the following Hamiltonian systems for $i=1, \ldots, d$ and $\sigma_i > 0$
$$
\begin{align}
\frac{d}{dt} x_i &= v_i \\
\frac{d}{dt} v_i &= \frac{x_i}{\sigma_i^2}
\end{align}
$...
- 129
0
votes
1
answer
23
views
Length of a regular arc
I have to prove that if two arc are (positive) equivalent, they have the same length.
Let $\alpha:[a,b]\to \mathbb{R}^n$ and $\beta:[c,d]\to \mathbb{R}^n$ two differentiable arcs and let $h:[a,b]\to [...
- 328
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votes
1
answer
51
views
Does limit exists or not?
I've been trying to solve with trajectories, but I haven't been able to find an answer.
$\lim_{(x,y)\to (0,0)} \frac{x^2y^4}{2x^2-2xy+2y^4}$
0
votes
1
answer
39
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Double integral $\int \int_R \sqrt{1-x^2-y^2} \mathrm{d}x\mathrm{d}y$ (from the surface integral $\int \int_S z^2 \mathrm{d}S$)
I have this double integral over a region $R\in \mathbb{R}^2$, which is the projection of the surface $S: \{x^2+y^2+z^2=1| \; x\geq0,y\geq0,z\geq0\}\in \mathbb{R}^3$ (located in the 1st quadrant).
I ...
- 179
5
votes
3
answers
305
views
Interpreting a notation in calculus of variations (differentiating with respect to a derivative)
Consider a functional $J[y]$ defined by:
$$J[y] = \int_a^b F(x, y, y') dx \tag{1}$$
Here, $F$ is a function that depends on the independent variable $x$, the function $y(x)$, and its derivative $$y' = ...
- 317
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votes
1
answer
35
views
A question about rudin 9.32 Why $PA=A$
9.32 Theorem: Suppose 𝑚,𝑛, are nonnegative integers, $𝑚≥𝑟$,$𝑛≥𝑟$,$𝐹$ is a $𝐶^{1}$ mapping of an open set 𝐸⊂$𝑅^{𝑛}$ into $𝑅^{𝑚}$, and $𝐹′(𝑥)$ has rank $r$ for every $𝑥∈𝐸$.
Fix $𝑎∈𝐸$...
1
vote
1
answer
46
views
Integrate a sum of trig function under absolute value
Let $n \in \mathbb{N}$, I'm trying to compute en explicit formula for the following integral:
$$
\operatorname{I}\left(n\right) = \int_{\left[0,2\pi\right]^{n}} \left\vert\cos\left(x_1\right) + \cdots ...
- 808
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votes
0
answers
51
views
Is the implicit function unique?
Does the equation $$xy^2+xz^3+\ln z=0$$ define the unique implicit function
$z=g(x,y)$ in the neighborhood of $(0,1)$? If yes, compute
$\dfrac{dz}{dx}(0,1)$ and $\dfrac{dz}{dy}(0,1).$
My attempt:
Let ...
- 436
0
votes
1
answer
45
views
Change of coordinates on $\nabla(h\circ\varphi^{-1})$ where $h,\varphi:\mathbb{R}^n\to\mathbb{R}^n$
Say I have a smooth vector-valued function $h:\mathbb{R}^n\to\mathbb{R}^n$ and a smooth diffeomorphism $\varphi:\mathbb{R}^n\to\mathbb{R}^n$. Consider the gradient of the composition $h\circ\varphi^{-...
- 1,714
0
votes
1
answer
41
views
functionally unfair coins
If the faces of a two-sided "coin" are parameterized by (convex in the direction of heads or tails, respectively) continuous functions $f_1(r,\theta)$ and $f_2(r,\theta)$ (of two variables: ...
- 90
1
vote
1
answer
70
views
Is this function differentiable? (Error in the notes?)
So I'm studying in my notes, and there is this "example" (not worked) where it's stated that the following functions is NOT differentiable at $(0, 0)$:
$$f(x, y) = \begin{cases} \frac{\sqrt{...
- 3,255
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vote
1
answer
89
views
Approximation of a non differentiable function
Can you please tell me if/where I am wrong about this?
Let $f: \mathbb{R}^2\to \mathbb{R}$ a function. Let $f$ be derivable at $\left( \frac{6}{5}, \frac{6}{5}\right)$ but not differentiable at that ...
- 3,255
0
votes
1
answer
24
views
Is there any way to compact the propagation of uncertainty formula in terms of vectors?
The uncertainty associated to $\xi=f(\mathbf x)$ with $\mathbf x\in\mathbf R^n$ is
$$\delta\xi=\sqrt{\sum_{i=1}^n\left(\frac{\partial \xi}{\partial x_i}\delta x_i\right)^2}$$
What I was wondering was ...
- 1,676
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votes
0
answers
26
views
The chain rule for partial derivatives
My goal is to understand a statement about partial derivatives made in the appendix of a textbook about manifolds. From the book:
Let $U \subseteq \mathbb{R}^n$ and $\tilde{U}\subseteq \mathbb{R}^m$ ...
- 1
0
votes
0
answers
33
views
Find an explicity formula for the inverse of the function $\textbf{f}$ given by $f_1=e^x\cos y$ and $f_2=e^x\sin y$ [duplicate]
Let $\textbf{f}$ be given by $f_1=e^x\cos y$ and $f_2=e^x\sin y$, and put $\textbf{a}=(0,\pi/3)$, $\textbf{b}=\textbf f(a)$. Find
an explicity formula for the inverse of $\textbf{f}$, denoted
$\textbf{...
- 2,749
1
vote
1
answer
71
views
Prove that $f$ doesn't attain its local maximum
Suppose that $f:\mathbb{R}^2\to\mathbb{R}^2$ is of class $C^3$ s.t
$$\Delta f=\dfrac{\partial ^2f}{\partial x^2}+\dfrac{\partial
^2f}{\partial y^2}>0\quad\forall (x,y)\in\mathbb{R}^2.$$ Prove that
...
- 436
0
votes
1
answer
49
views
Why isn't the integral of the total differential of a function equal to the function? [duplicate]
For example let $w = 2x^3y$
then $$dw = \frac{\partial w}{\partial x}dx + \frac{\partial w}{\partial y}dy $$
$$\implies dw = 6x^2y\hspace{1.5mm} dx + 2x^3 \hspace{1.5mm} dy$$
Then why isn't it that $$...
- 11
0
votes
2
answers
29
views
Composition of a $L^2$ function with a $C^1$ monotone function (with some additional conditions) is again $L^2$
I don't know why this problem is giving me so much trouble but for whatever reason I cannot do this, maybe it's just too late in the day. Take $U \subseteq \mathbb R^n$ be open and bounded, $f: U \to \...
- 1,373
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votes
0
answers
17
views
A second-order sufficient condition with a local condition for extreme value [closed]
Assume that f and g are twice continuously differentiable in a neighborhood of (x0, y0), g(x0, y0) = 0,
(∇g)(x0, y0) ̸= 0 and (∇L)(x0, y0, λ0) = 0 for some λ0 ∈ R, where L(x, y, λ) = f(x, y)−λ * g (x, ...
2
votes
0
answers
21
views
Stokes theorem to calculate line integral
Let $\gamma$ be the intersection between $z=x^2+y^2$ and the plane $z=1+2x$. Calculate the work done by the field $F=(0,x,-y)$ when the curve $\gamma$ traverses on lap in positive direction seen from ...
- 765
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votes
0
answers
14
views
Why the limit by the level curve path c is always equals c? [closed]
i have a very begginer question, but i can't find the reason why the limit of any function of two variables when we take the level curve $f(x, y) = c$ path is equal c.
I am thinking of this because, ...
- 1
0
votes
0
answers
40
views
Calculus Identities
I am trying to write an expression to $\partial_t \|\nabla u\|_{L^p(\Omega)}^p.$ Here $\Omega$ is a smooth domain, the function $u$ has no regularity problems (you can take it smooth) and the normal ...
- 343
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0
answers
88
views
Inverse function theorem in the case of $\det=0$
Let $f:\mathbb{R}^2\to\mathbb{R}^2$ be defined by $$f(x,y)=(x^2-y^2,
2xy).$$
a) Does $f$ have local inverse function at $(0,0)$?
b) Find the domain of $f$.
My attempt:
a) I just prove $f$ has local ...
- 1,255
2
votes
2
answers
58
views
A problem using Chain Rule theorem
Let $F(u,v)$ be of class $C^1(\mathbb{R}^2)$ and $z=z(x,y)$ be the
function defined by the equation
$$F\left(x+\dfrac{z}{y},y+\dfrac{z}{x}\right)=0.$$ Prove that the
function $z(x,y)$ satisfies the ...
- 1,255
-1
votes
1
answer
38
views
Question about directional derivatives [closed]
The temperature of a $10\times 10$ inch hot plate is given by $T(x,y) = 100-x^2 - 2y^2$ at $x$ units to the right of the lower left and $y$ up from the lower left.
If a bug on the plate can survive ...
1
vote
0
answers
52
views
How to find the maximum and minimum values, if any, of $G(x,y)=\frac{\sin(\pi \sqrt{x^2+y^2})}{\pi \sqrt{x^2+y^2}}?$
I have been trying to find the critical points of the function $G(x,y) = \frac{\sin(\pi \sqrt{x^2+y^2})}{\pi \sqrt{x^2+y^2}}$. The partial derivatives are
$$G_{x}(x,y)=\frac{x}{\pi (x^2+y^2)^{3/2}} \...
- 133
0
votes
1
answer
39
views
A point in a set is interior point iff
Definition of interior point: $x$ is interior point of $X\subset\mathbb{R}^n$ if there exists $\varepsilon >0$ such that $B(x,\epsilon )\subset X$.
Let $A\subset\mathbb{R}^n$. $a\in A$ is an ...
- 1,296
0
votes
0
answers
33
views
What does it mean for an inclusion map to be smooth?
Sorry for the elementary question, but I do not understand what does it mean for an inclusion map to be "smooth"? My understanding is that an inclusion map will send an element $x$ into ...
- 161
0
votes
0
answers
21
views
Conservative vector fields: curves and path [closed]
Suppose a vector field $F$ has the property that
$$
\int_C F =0
$$
for all curves $C$ of class $C^1$. Is it true that the same property holds if we change $C$ for a path (yuxtaposition of $C^1$ curves,...
- 499
0
votes
1
answer
104
views
Why $W$ is open in baby Rudin 2.28
9.28 Theorem: Let $f$ be a r $\mathscr{C}'$ -mapping of an open set $E$ $ \subset \mathbb{R}^{n+m}$ into $\mathbb{R}^n$, such
that $f(a, b) = 0$ for some point $(a, b) \in E$.
Put $A = f'(a, b)$ and ...
-2
votes
0
answers
21
views
Determining the Constant for Maximum Rate of Increase in a Multivariable Function Along a Given Vector [closed]
Let $f (x, y) = x^3 y − x y^2+ c x^2$ where $c$ is a constant. Find $c$ if $f$ increases fastest at the point $P_0 (3, 2)$ in the direction of the vector $\vec{v} = 2 \hat{i} + 5 \hat{j}$.
- 1
1
vote
0
answers
39
views
Generalization of mean value theorem?
For differentiable function $f: \mathbb{R} \to \mathbb{R}$ the mean value theorem states:
given $a, b$ there exists $c$ such that
$$
f(b) - f(a) = f'(c) (b-a).
$$
Is there something similar for $f: \...
- 2,917
0
votes
1
answer
25
views
Conditions for Symmetry of mixed partials
I am studying symmetry of mixed partial derivatives in multivariate calculus(Clairaut's Theorem). Wikipedia says that differentiability of first order partials implies symmetry, to which I have found ...
- 3
2
votes
1
answer
31
views
If $F$ attains its local minimum at $x=0$ then $F(x)-F(0)\geq\xi x$
Let $F:\mathbb{R}\to\mathbb{R}$ be the convex mapping. Prove that
there exists derivatives $F'(0+), F'(0-)$ and if $F$ attains its local minimum at $x=0$ then $$F(x)-F(0)\geq \xi x,\quad\forall x,\xi\...
- 436
0
votes
1
answer
42
views
Deal with discontinuity in double integrals
Let $f(x,y)= \frac{x^2+y^2}{x^2}$. Consider
$$ \int_D f(x,y)$$
with $D=\{ (x,y): 0\leq y\leq x, x^2+y^2\leq 1 \}$
The function is unbounded in $D$, due to the denominator. I checked that cannot be ...
- 499
-5
votes
0
answers
38
views
Suppose the temperature at (x,y,z) is given by M=xyz. In what direction could you go from (1,1,1) to keep the same temperature? [closed]
I’ve got this problem (which has been posed but I did not understand the answer to). I do not fully understand gradients and directional derivatives so could anyone point me in the right direction for ...