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).

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8 views

Find the line of intersection of the planes

The planes are x+2y+3z=1 and x-y+z=1. My guess would be to set them equal to each other, since they are both equal to 1, we could write that as x+2y+3z=x-y+z. This simplifies to 3y+2z=0, it doesn't ...
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15 views

Second order differential equation, orthogonality

A temperature field T(x, t) is determined by the following governing equation: $$\frac 1\alpha\frac {dT}{dt} = \frac {d^2T}{dx^2}$$ (Eq 1) T(x,t) can be expressed as a form of expansion of T(x,t) = ...
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0answers
15 views

Vanishing eigenvalues of Jacobian

Let $f: \mathbb{R^2}\to \mathbb{R^2}$ be a Schwartz function. If the eigenvalues of $Df$ all vanish, must $f$ be constant? It is clear that the condition on $Df$ forces $\nabla \cdot f =\text{Tr } ...
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1answer
17 views

Second derivative expression

I have $f:\mathbb R^n\to \mathbb R$ and $\gamma:\mathbb R \to \mathbb R^n$, which are both $\mathrm C^2$. Considering $g=f\circ \gamma$, how could I express $g''$, second derivative of $g$ in terms of ...
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3answers
55 views

Books on differential geometry in the cases $n=2$ and $n=3$

I'm interested in learning the differential geometry of standard, "physical" space, that is $\mathbb R^2$ and $\mathbb R^3$. The sort of problems that were studied in the 18th and 19th century... ...
3
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1answer
33 views

Evaluate the integral $\iiint\limits_E x^2 \,\, \mathrm{d}V$

Where E is the region bounded by the xz-plane and the hemispheres $y=\sqrt{9-x^2-z^2}$ and $y=\sqrt{16-x^2-z^2}$. This is an exercise from my professor guide. What I tried so far: These exercise ...
3
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1answer
51 views

Find vector field given curl

I have an equation $\nabla \times \vec{B} = \mu_{0}\vec{J}$, where $\vec{J} = \left\langle f(x,y), g(x,y), 0 \right\rangle$ and need to solve for $\vec{B}$. I've looked elsewhere on here for how to ...
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1answer
51 views

Evaluate line integral without parameterizarion

It's been brought to my attention that line/surface integrals and integrals of differential forms in general can be evaluated without introducing a parameterization, however I haven't been able to ...
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0answers
11 views

Looking for a 'Flow-Free' Proof of the Rectification Theorem.

I am looking for a proof of the following theorem: Rectification Theorem: Let $\mathbf V:\mathbf R^n\to\mathbf R^n$ be a smooth function and $\mathbf p\in \mathbf R^n$ be such that $V(\mathbf ...
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0answers
22 views

Calculating Surface Area using double integrals [on hold]

Double integrals can be used to find the surface area of a surface defined over a region. Use the formula for calculating surface area to find the surface area of $z = \frac{x^ 3}{2} + \frac{y^3}{2}$ in ...
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1answer
14 views

Proofs involving Gradient

I need to prove the following $$\bigtriangledown ||\textbf{v}|| = \dfrac{1}{||\textbf{v}||} \textbf{v}$$ and $$\bigtriangledown (||\textbf{v}||^2) = 2\textbf{v}$$ I'm very confused about how I ...
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0answers
19 views

Partitioning a non bijective to bijective function

I would like to ask about a particular problem. I have the following equations Z= X^2 + Y^2 and W=X/Y Graphically I´d have a circumference with its center at (0,0) (X^2+Y^2) with a radius of ...
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0answers
24 views

Integrating differential forms over a box

I've only ever seen examples of integrating a differential form over a curve C involving defining a parameterization. I have seen people integrate 1 forms over a box without defining a ...
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1answer
27 views

Partial differential chain rule with designation of some variables being held constant

If a function $f$ can be expressed in two different coordinate systems say $(x,y)$ and $(\bar x,\bar y)$, how would one take the partial differential of $f$ with respect to $\bar x$ with $\bar y$ held ...
2
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1answer
22 views

Integration against divergence free vector fields

Let $\chi:\Omega\to \mathbb{R}^n$ be a vector field on a bounded, smooth domain $\Omega \subset \mathbb{R}^n$. Assume that for any divergence free vector field $\eta:\Omega \to \mathbb{R}^n$ we have ...
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0answers
9 views

CDF of smallest eigenvalue of non-central Wishart matrix - how to evaluate the integral.

Does anybody know how to derive the distribution of the smallest root of a non-central Wishart matrix? I have got an integral expression that would give me the desired answer but cannot solve the ...
2
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1answer
26 views
+50

Calculating $\text{D}g$ of $g(x,y) = \int_\frac1x^1\frac1t\exp(t^3x^2y)\text{d}t$

Let $g:(1,\infty)^2\to\mathbb{R}$ be given by $$g(x,y) = \int_\frac1x^1\frac1t\exp(t^3x^2y)\text{d}t.$$ How can I calculate $\text{D}g$ using parameter-dependent integrals?
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0answers
16 views

A question on Lagrange multipliers

The state of Megalomania occupies the region $x^4 + y^4 \leq 30,000.$ The altitude at the point $(x,y)$ is $\frac{1}{8}xy+200x$ meters above sea level. Where are the highest and lowest points in the ...
2
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1answer
47 views

Evaluating $\lim\limits_{(x,y)\rightarrow(1,1)} \frac {\sin(x) - \sin (y)} {x-y}$

I am taking a calculus exam in less than one week, and I've stumbled upon this expression. $$\lim\limits_{(x,y)\rightarrow(1,1)} \frac {\sin(x) - \sin (y)} {x-y}$$ Which I know to be cos(1), but ...
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0answers
10 views

Regular Value Theorem Using Implicit Function Theorem in Calculus.

I want to prove the follwoing: THEOREM. (Regular Value Theorem.) Let $f:\mathbf R^n\to\mathbf R^m$ be a smooth function and $\mathbf a\in\mathbf R^n$ be a regular point of $f$. Let ...
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0answers
12 views

If f(x,y)=g(u,v) and u(x,y), v(x,y); is $\frac{\partial f}{\partial x} = \frac{\partial g}{\partial x}$ true?

It's related to a question which I could prove assuming it to be true. Now I want to know if it's true in general.
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1answer
18 views

Vector calculus identities

Let $f$ be scalar potential for the vector field $\underline u $ (i.e $\underline u = -\underline \nabla f$). Prove that the vector field $$ \underline r \wedge \underline u $$ has magnetic ...
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1answer
33 views

Gradient of real part = real part of gradient?

Suppose f(x,y,z) maps $\mathbb{R}^3\rightarrow\mathbb{C}^1$. That is, it takes in three real numbers and spits out a complex number. Does the following always hold: $$\vec\nabla ...
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1answer
16 views

why $f|_{U}$ is immersion for some open set containing $a$

Its written that if $f$ is immersion at $a$ then $f|_{U}$ is immersion for some open set containing $a$. I don't understand why its happening.. can one please explain ?
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2answers
54 views

Find the volume below $\sqrt{x}+\sqrt{y}+\sqrt{z}=1$ in the first quadrant

I understand that we have to use transformation $$x = u^2, y = v^2, z = w^2$$ but I cannot figure out the limits. I just need a rough sketch of how to approach this. Could anyone give me some ideas?
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2answers
53 views

Find the smallest value of $f(x, y, z)$

Find the smallest value of $f(x, y, z) = \sqrt{x^2 + 1} + \sqrt{(y - x)^2 + 4} +\sqrt{(z - y)^2 + 1} + \sqrt{(10 - z)^2 + 9}$ I found this question while looking from some exam papers and have no ...
2
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2answers
48 views

What is the order of the PDE $\newcommand\pp\partial\frac{\pp^2u}{\pp x^2}+\frac{\pp^3u}{\pp x^2 \pp y}+\frac{\pp^2u}{\pp^2y}=xy\frac{\pp u}{\pp x}$? [on hold]

The order of the differential equation $$\frac{\partial^2 u}{\partial x^2}+\frac{\partial^3 u}{\partial x^2 \partial y}+\frac{\partial^2 u}{\partial^2 y}=xy\frac{\partial u}{\partial x}$$ is ...
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2answers
79 views

2 examples to try to understand partials derivatives and deriviability

To prove that a functions has partial derivatives every partial has to exist, and every partial exist only if the limit of definition of partial exist. Is this right? Then if partials exist ,and the ...
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0answers
16 views

Deduction of the equation of continuity in one dimension

I need the deduction of the continuity equation in one dimension usig the result: $$\frac{\partial}{\partial t}\int^{b(t)}_{a(t)} \rho(x,t)dx=\rho(x,t)b'(t)-\rho(x,t)a'(t)+\int^{b(t)}_{a(t)} ...
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0answers
14 views

parametrization of intersecting level curves in neighborhood of given point

Let $f(x,y,z) = yarctan(x) +z^2,g(x,y,z) = xy^2 + xyz + z $ and let $\gamma$ be the intersection curve between the surfaces $f(x,y,z) = 1$ and $g(x,y,z) = 1$ Show that $\gamma$ can be ...
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0answers
19 views

Fréchet normal cone

Given $x\in \Omega(\subset X)$ (X: Banach space) and $\varepsilon\geq 0$, the set of $\varepsilon-$normals to $\Omega$ at $x$ by \begin{align} \widehat N_\varepsilon(x;\Omega):=\left\{x^*\in X^*\mid ...
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3answers
23 views

Check if two vector equations of parametric surfaces are equivalent

Give the vector equation of the plane through these lines: $\begin{pmatrix}x\\y\\z\end{pmatrix}=\begin{pmatrix}4\\1\\1\end{pmatrix}+\lambda\cdot\begin{pmatrix}0\\2\\1\end{pmatrix}\,\,\,$ and ...
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1answer
30 views

Continuity of the maximum of a function in two variables

The function $f( x, y)$ is continuous on $x\in [a,b]$, $y\in [a,b]$. Is the function $g(x) = \max_{y} f( x, y)$ continuous on $x\in [a,b]$?
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0answers
29 views

Solving the Telegraph Equation using Partial Differential Equations and Sturm-Liouville theory

I've been asked to do the following question, and I've got through the brunt of it (so this is going to be a rather long question...), but I'm just having a bit of trouble applying Sturm-Liouville ...
0
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1answer
19 views

Volume of a solid in spherical coordinates

How might we find the volume of the solid whose surface is $\rho = \sin{\phi}^{1/3}$? Of course, the obvious way to proceed is to write the triple integral $$\int_V dV$$ taking of course $dV = ...
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1answer
23 views

How can I numerically evaluate the total derivative of a multivariate function?

I think I understand now the intuitive reasoning behind the total derivative of a multivariate function $z = z(x, y)$, which is $$ dz = \frac{\partial{z}}{\partial{x}}dx + ...
1
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1answer
28 views

Confusion about the Total Derivative

I just started multivariable calculus a little while ago and I'm confused about the concept of a total derivative of some function $z = z(x, y)$. I was taught that $dz = \frac{\partial z}{\partial ...
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2answers
30 views

no diffeomorphism from $\mathbb{R}^2 \to \mathbb{R}^3$

Show there is exist no diffeomorphism from $\mathbb{R}^2 \to \mathbb{R}^3$ PS: Don't say $\mathbb{R}^2,\mathbb{R}^3$ aren't homeomorphic, I need explanation without using topology
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0answers
23 views

Normal Vector Affecting The Divergence Theorem

$\newcommand{\Div}{\operatorname{Div}}$I'm going to use an example to explain what I'm trying to ask. Let $T =\{(x,y,z): x^2+y^2=z^2, 0\leq z\leq3\}$, I'm asked to calculate $\iint_T ...
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17 views

Firm non-expansiveness in the context of proximal operators

$\newcommand{\prox}{\operatorname{prox}}$ Probably the most remarkable property of the proximal operator is the fixed point property: The point $x^*$ minimizes $f$ if and only if $x^* = \prox_f(x^*) ...
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1answer
13 views

line integrals and partial derivatives statement (Green's theorem application)

Let $P(x,y),Q(x,y)$ be $C^1$ functions of $\mathbb R^2$, prove that the following statements are equivalent: (1) $P_x-Q_y=0$ and $P_y+Q_x=0$ (2) For every simple closed curve $C$, it is satisfied ...
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1answer
26 views

geometric interpretation of the norm: $\|\vec x\|={(|x_1|+|x_2|)\over 3}+{2\max(|x_1|,|x_2|)\over 3}$

Let $p:\mathbb R^2 \to \mathbb R$ be a norm so that $$\|\vec x\|={(|x_1|+|x_2|)\over 3}+{2\max(|x_1|,|x_2|)\over 3}$$= $${{\|\vec x\|_1\over 3}}+{2\|\vec x\|_\infty\over 3}$$ The thing is that I need ...
4
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1answer
37 views

Mixed partial derivatives are different

Let $f: \Bbb R^2 \to \Bbb R$ be defined as $$f(x) = \left\{ \begin{matrix} x_1^2 \operatorname{arctan} \left( \frac{x_2}{x_1} \right) - x_2^2 \operatorname{arctan} \left( \frac{x_1}{x_2} \right), ...
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1answer
41 views

Calculating multi-variable limit.

I am struggling to find a way to approach this limit $$\lim_{(x,y)\to(0,0)}\frac{\sin(x^2y+x^2y^3)}{x^2+y^2}$$ I would greatly appriciate if You could explain to me how to solve it or at least show ...
2
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1answer
20 views

continuity single and multivariable function simple question

Why $$f(x,y) =\begin{cases} \frac{xy^2}{x^2 +y^2} \mbox{ for } (x,y)\neq (0,0) \\ 0 \mbox{ for } (x,y)= (0,0)\end{cases}$$ is continuous and $$f(x) =\begin{cases} 2 \mbox{ for } 0>=x>10 \\ 5 ...
0
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1answer
20 views

continuity single variable function and multivariable funtion and its parcial derivatives

Is f(x)=1/x discontinuous at point x=0 or not since its domain is x>0 and x<0? And what about f(x,y)=$\frac{xy^2}{x^2+y^2}$ continuity? And Df(x,y) exist or parcial derivatives are ...
2
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1answer
23 views

Surface Integral over a sphere

Suppose $f(x,y,z)=g\left(\sqrt{x^2+y^2+z^2}\right)$, where $g$ is a function of one variable such that $g(2)=-5$. Evaluate $$\iint_S f ~dS,$$where $S$ is the sphere $x^2+y^2+z^2=4$. Now, I ...
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1answer
28 views

Optimization with a constraint given by a differential equation

I have the following differential equation $$\ddot\theta(t) = -k\sin{\omega t}\sin{\theta(t)} \quad \text{where} \quad \theta(0)=\theta_0, \dot\theta(0)=v_0$$ where $\omega$ is a known constant and ...
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0answers
60 views

Can a set in $\mathbb{R}^2$ be closed but unbounded?

Today I read "on a closed, bounded set $D$". How can a set be closed but not bounded?
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0answers
28 views

Sufficient Conditions for Multivariate Decreasing Function

I found the following helpful theorem concerning decreasing functions but it's only valid for $\varphi:\mathbb{R}\rightarrow \mathbb{R}$, I'd like to know if it can be extended to the ...