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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Basic Partial Differential Chain Rule

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 ...
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3answers
61 views

If a plane intersects a regular surface at exactly one point, then it is the tangent plane

Question Let a regular surface, $S$, intersect a plane, $P$, at only one point, $p_0 = (x_0, y_0, z_0)$ in $\mathbb{R}^3$. Show that the plane coincides with the tangent plane to the surface at ...
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7 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 ...
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0answers
11 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
7 views

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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15 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 ...
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1answer
40 views
+50

Rewriting a continuously differentiable function

I have the following $i$-th regressor function: $\phi_i(x)$ in which $x$ is a vector with elements $x_1, \ldots, x_n$. I cite from an article: Let $e_i = \hat{x}_i - x_i$ and note that, since ...
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1answer
38 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
8 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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24 views

how to check partials derivatives are continuity?

If a function has partials how to check that the partials are continuous ? And how on a piecewise function? If f is define as$$f(x,y) =\begin{cases} x^2+2x+5y+10 & \text{ for } (x,y)>= (0,0) ...
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0answers
13 views

how to prove a multivariable function is differential? is another procedure

How to prove a multivariable function is derivable? Is to see if a function has some points of discontinuity by inspection and check on every point of possible discontinuity using the definition of ...
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2answers
383 views

How to find extrema on a triangle

Let $T\subset\mathbb{R}^2$ be the (closed) triangle bounded by the lines $x+y=4$, $x\ge-1$ and $y\ge-1$. I want to find and classify all the extrema of the function $f(x,y)=-x^2y(x+y-2)$ on the ...
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1answer
15 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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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
32 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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2answers
47 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 ...
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2answers
47 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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0answers
62 views

How to prove partials derivatives are continuous?

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 you have partials and ...
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0answers
14 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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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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0answers
12 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
17 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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1answer
26 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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27 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 ...
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36 views

Partial derivatives of a piecewise defined function

If a function f as $$f(x,y) =\begin{cases} x^2+2x+5y+10 & \text{ for } (x,y)\neq (0,0) \\ y^2+2y+x+10 & \text{ for } (x,y)=(0,0) \end{cases}$$ Is it true that $$f^\prime_x(x,y) ...
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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
178 views

Absolute extrema

Find the absolute extrema of the function $f(x,y)=2xy-x-y$ over the region of the xy-plane bounded by the parabola $y=x^2$ and the line $y=4$ I was wondering if I needed to use Lagrange multipliers to ...
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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 + ...
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3answers
134 views

How to define intrinsic curvature?

I have been exploring differential geometry slightly... And i'm trying to grasp how to define intrinsic curvature, from a visual/geometric viewpoint... One formulation I got was that Intrinsic ...
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1answer
26 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
3k views

Gradient of a dot product

The wikipedia formula for the gradient of a dot product is given as $$\nabla(a\cdot b) = (a\cdot\nabla)b +(b\cdot \nabla)a + a\times(\nabla\times b)+ b\times (\nabla \times a)$$ However, I also ...
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1answer
56 views

minimum and maximum value of 2-variable function

How do you find min,max values of $f(x,y)=xye^{xy-4y}$ when $-4\le x \le 1$ and when $-1\le y \le 4$. min and max when $x=-4$ and $-1\le y \le 4$ or when $x=1$ and $-1\le y \le 4$ and the global ...
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2answers
29 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
22 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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0answers
15 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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2answers
659 views

Shortest distance between two curves

Let $C_1= \{ (x, y) \in \mathrm{R}^2 : y = x^2 +1 \}$ and $C_2= \{ (x, y) \in \mathrm{R}^2 : x = y^2 +1 \}$, find the points which minimize distance between $C_1$ and $C_2$. What I tried is: we know ...
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1answer
61 views

Minima maxima for a 3 variable function on a whole Critical line ( and not a point)

Say if i have a function $f(x,y,z)= xyz(16-x-y-2z)$ and i am looking for maxima and minima for it. With a quick calculation and after demanding that the $\nabla f = 0$ We get that the critical ...
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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
24 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 ...
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1answer
40 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 ...
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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
21 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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2answers
67 views

When is $\min_{x\in X,y\in Y}f(x,y)=\min_{x\in X}(\min_{y\in Y}f(x,y))$?

When is $$ \min_{x\in X,y\in Y}f(x,y)=\min_{x\in X}(\min_{y\in Y}f(x,y))? $$
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2answers
43 views

Find the work done by the force field in moving the particle from one point to another

Find work done by the force field F in moving the particle from $(-1, 1)$ to $(3, 2)$ This sounds good till we are given that $\textbf{F} = \dfrac{2x}{y}\textbf{ i }- \dfrac{x^2}{y^2}\textbf{ j }$ ...