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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31 views

Double integral over complicated region

Suppose we wanted to compute $\iint\frac {1}{1 + x^2 + y^2} dxdy$ over the region $\frac {(x-1)^2}4 + \frac {(y+2)^2}9 \leqslant 1$. It gets quite hairy if we use elliptical polar coordinates i.e. ...
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1answer
37 views

Local minimum and gradient [duplicate]

But the proof here below is specially elegant. Is there any function $f$ such that $f$ has a local minimum at $x$ but $\nabla f(x) \neq 0$? Only assumption on $f$ is that it has to be differentiable ...
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2answers
36 views

Continuity of a multivariable function with “parts”

I'm trying to solve if $f$ is continuous: $$ f(x,y) = \begin{cases} x^3 + y^3 &\text{if }y>0 \\ x^2 &\text{if }y ≤ 0 \end{cases} $$ I have seen that $$\lim_{(x,y) \to (0,0)} ...
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1answer
18 views

Finding the image of a region transformed by a mapping

The only examples I've found are either very complicated, or state the transformation like y=g(u,v) x=f(u,v), whereas this question states u and v in terms of x and y. I'm not sure how to get ...
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1answer
17 views

Find an equation for the plane

Here is the whole question. Find an equation of the plane that passes through the points $P(1,0,-1)$ and $Q(2,1,0)$ and is parallel to the line of intersection of the planes $x+y+z=5$ and $x+y-z=1$. ...
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27 views

Just a curious question. What type of mathematics do we need to deal with a vector valued- function that depends on multiple vectors?

We have a scalar valued function that depends on multiple variables (takes in a vector), what about a vector valued function that depends on multiple vectors??
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1answer
27 views

Particular solution of reducible to homogeneous equation

Verify that $y=x-5$ is a particular solution of the equation $$\frac{dy}{dx} = \frac{2y+6}{x+y+1}\ .$$ This is when $y'=1$ but this is not given as a condition in the question. How would you write the ...
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1answer
17 views

Meaning of the Notation $\mathcal{F}[X,F(X)]$

I came across the following theorem. What is the meaning of the notation $\mathcal{F}[X,F(X)]$? Thanks, Jay.
2
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1answer
62 views

Calculating a multivariate probability density - how to invert the function?

This is an example from a lecture, however it was presented without proof, so I'm trying to find a way to calculate the PDF for the given condition: $X_1$ and $X_2$ are two independent random ...
0
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1answer
34 views

Is there a formal proof for this theorem??

There is a theorem in the book Advanced Calculus by Wilfred Kaplan which states the following: The differential formula : $$ dz = \frac {\partial z}{\partial x} dx + \frac{\partial z} {\partial y}dy ...
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1answer
233 views

gradient of vector 2-norm

I have a function $f(\Theta) = \frac{1}{2N}\| y-\mathcal{X}(\Theta)\|_2^2$. Matrix $\Theta\in\mathbb{R}^{m_1\times m_2}$, $y=[y_1,\cdots,y_N]^T\in\mathbb{R}^N$ is the observation vector, and we use ...
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3answers
50 views

Find $\int_0^4\int_{0}^{4}xy \sqrt{1+x^2+y^2} \,dy\, dx $

I am having a tough time figuring this one out. Any help will be appreciated. do we have to approximate, or can we actually find it
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1answer
29 views

Finding volume between plane and paraboloid

Find the volume between bounded by $z=4$ and $z=x^2+y^2$.(Answer: $8\pi$) I thouhg using dievergence theorm ($\iint_KdivFdxdydz=\iint_SF\cdot\hat{n}dS$) for $\vec{F}=\big(\frac x 2,\frac y ...
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1answer
24 views

Special form of the Divergence theorem

It states in my notes relating to the derivation of the euler equations that a 'special form' of the divergence theorem is: $\iint{\phi\hat{n}}\space{dS}=\iiint\nabla\phi\space{dV}$ with $\phi$ a ...
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0answers
21 views

Implicit Function Theorem (Two Variables)

While going through a Mathematica Handout (from the web: Link), I came across the following (last page of the PDF copy): How did the author derive the equation $g(x,t)=x-h(t)$ using the conditions ...
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2answers
66 views

Typed version of Newton's Principia Mathematica

I need a typed pdf version of Newton's Principia. Is it available for free online? And I also need the proof of universal law of gravity and the elliptical orbits of planets(If there's no typed ...
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2answers
31 views

Double integration:$ \int_0^a \int_0^b e^{max(b^2x^2,a^2y^2)}dydx $

I would be grateful for a little help if someone could help me solve a problem in my textbook. The question is, evaluate $ \int_0^a \int_0^b e^{max(b^2x^2,a^2y^2)}dydx $, where $a,b$ are positive ...
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1answer
19 views

Find the point of intersection of plan and parabaloid

Find the point of intersection of the plane $x+2y+z=10$ and the parabaloid $z=x^2+y^2$ that is closest to the origin. Do this by minimizing the distance squared from the origin: ...
0
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1answer
28 views

Find the absolute extrema by Lagrange multipliers

Find the absolute extrema for the function $g(x,y)=e^{x^2}-y^2$ on the unit disk $D$ given by: $D=\{(x,y)|x^2+y^2\le1\}$. Do this by first finding all critical points of and classifying them, then ...
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11 views

find the absolute extrema

Optimize the function f(x,y)=x^2y on the elliptical cylinder x^2+2y^2 =< 6 using Lagrange Multipliers well, from what I know that I have to find the gradient then to set it equal to zero but i'm ...
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19 views

a problem on differentiable map

Let $T:\mathbb{R}^n \to \mathbb{R}^m$ be a map. Let $a \in \mathbb{R}^n$. We define a map $g:\mathbb{R}^n \to L(\mathbb{R}^n,\mathbb{R}^m)$ such that $g(a)=DT(a)$ where $D$ denote the differential of ...
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1answer
107 views

Find maximum of $P$

Let $$P = \frac{{{x^2}}}{{{x^2} + yz + x + 1}} + \frac{{y + z}}{{x + y + z + 1}} - \frac{{1 + yz}}{9}.$$ Find maximum of $P$ where $x, y,z$ are nonnegative real numbers such that ${x^2} + {y^2} + ...
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3answers
55 views

What is the range of $f(x,y)=e^{-(x^2+y^2)}$

I know the domain is $\mathbb{R^2}$. Is the range $\mathbb{R}$?
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2answers
39 views

Integrating a differential form inside a cylinder

Let S be cylinder given by $x^2+y^2=1$ between $z=1$ and $z=3.$ For $\varphi=e^xdx\wedge dy+ ydz\wedge dx+xdy\wedge dz$ find $\int_S\varphi$. I managed to finish the problem, but I'm getting ...
4
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3answers
115 views

Why can't we usually speak of partial derivatives if the domain is not open?

In the book by Guillemin & Pollack they define a function $f$ from an open subset $U\subset R^n$ to $R^m$ to be smooth if it has continuous partial derivatives of all orders. Then they say ...
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1answer
18 views

Compute double integral on polar coordinates, find $r(\phi)$

I have the function $f(x,y)=y^2-2x^2y+6x^3-3xy+2y-6x$ and the region $\{y\geq 2x^2-2, y\leq 3x\}$. The region is: To compute the integral in cartesian coordinates: ...
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1answer
21 views

The acceleration in terms of the eulerian velocity

I'm having trouble in deriving the the acceleration in terms of the eulerian velocity. How do I apply the chain rule for partial derivatives to achieve the result highlighted in green?
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22 views

Average value of function over sphere

Here is another qual problem. Suppose $f:\mathbb{R}^3 \mapsto \mathbb{R}$ is $C^2$, and define the (scaled) average function $A(r)=\int_{S^2} f(rn) \:d\sigma(n)$, where $\sigma(n)$ is the usual ...
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1answer
30 views

Question involving double integral

How can I calculate the integral $$\iint_D \frac{\partial}{\partial y}\frac{y}{(x^2+y^2)^2} \, dx \, dy$$ where $D=\left\{ (x,y) \in \mathbb{R}^2 \mid 1\leq x^2+y^2 \leq 4 \right\}$ ? Thanks !
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1answer
12 views

Representing a triple integral in a different order of integration

I am given with the following question: A) $V_1 = \{ x^2 + y^2 \leq 4 , 0\leq z\leq 3 \sqrt{x^2 + y^2 } , x\geq 0 \} $ , and I need to represent the triple integral $\int \int \int_{V_1} f(z) dxdydz$ ...
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2answers
50 views

Find the gradient of $\frac{x}{x-y}$

It seems simple on the face of it, but I cannot figure out how to actually do this. I know that you have to find the partial with respect to $x$ and also with respect to $y$, but that's where I get ...
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1answer
35 views

Changing order of integration in cylindrical coordinates

I'm having a problem in changing order of integration in triple integration, in cylindrical coordinates. I would be grateful for a little help.The question is: Let D be the region bounded below by ...
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1answer
214 views

determine the resultant speed of the plane

a plane flying due east at 200 km/h encounters a 40-km/h wind blowing in the north-east direction. the resultant velocity of the plane is the vector sum v = $v_1$ + $v_2$, where $v_1$ is the velocity ...
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1answer
46 views

evaluating surface integral with divergence theory

If I have to calculate the surface integral of $\iint_S A \cdot n\ \mathrm {ds}$ where $A= 3zi-2xj+5x^2zk$ and $S$ is the surface of the cylinder $x^2+y^2=4$ and lying between $z=0$ and $z=4$ in the ...
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2answers
266 views

Boundary under transformation of a closed curve from $R^2\to R^3$

Consider some mapping $\phi: R_{uv} \to S\subset \mathbb{R}^3$ where $R_{uv}\subset \mathbb{R}^2$ and such that it is a simply connected region. We call the boundary of the surface (which we ...
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1answer
20 views

Surface integral using Stokes' theorem

$$ \int_\Gamma y\,dx+z\,dy+x\,dz $$ when $\Gamma$ $= \{ (x,y,z): x^2+y^2+z^2=9\}$ $\cap$ $x+y+z=0$ There's a theorem that states: $\int_S(\nabla \times \vec F)\cdot d \vec S$= $\int_S(\nabla \times ...
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0answers
15 views

About integration limits over region

I need to integrate $f(x,y)=y^2-2x^2y+6x^3-3xy+2y-6x$ over $\{y\geq 2x^2-2, y\leq 3x\}$ Im in doubt if it can be computed in just "one step" as: ...
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0answers
30 views

Reference Request: Fubini's theorem for non-negative functions

I have never seen this (1st page) formulation of Fubini's theorem in the literature. Does anyone know where I can find it? In every calculus book (e.g. Apostol, Courant, etc.) I looked, the authors ...
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2answers
18 views

Volume under surface

What is the volume under the surface $z = f(x,y) = x^4 + xy + y^3$ over the rectangle $R = [1,2] \times [0,2]$. I solved the double integral by integrating with respect to $x$ and then with respect ...
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1answer
26 views

Understanding arguments to functions in $\mathbb R^n$

Example of two theorems I have problems with: Mean value theorem: $U\subseteq\mathbb R^n$ open, $f:U\to\mathbb R^m$continuously differentiable, $x\in U$, $\xi\in\mathbb R^n$ such that $x+t\xi\in U$ ...
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15 views

Question about the formal justification of nondimensionalization

Assume I have the following (very simple) problem: $\frac{\partial f}{\partial \theta} =0 $ and I want to make a change of variables to make it nondimensional. So, I can write: $ F = \frac{f}{f^*}$ ...
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63 views

Second degree partial differential equation with variable-change

Edit: @Etienne mentioned that I did a typo, writing $u_y' = -xye^{-y}$ instead of $u_y' = -xe^{-y}$. I've corrected that in the calculations and now it's closer to being correct! Though I still miss ...
0
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2answers
34 views

Multivariable limit with parameter

I'm trying to solve a limit: $$\lim_{(x,y)\to(0,0)} \frac{(yx^n)}{(x^6+y^2 )} \forall n\ge4$$ i dont know how can i start to solve it, can someone help me? thanks
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2answers
35 views

Variable substitution in second order PDE

Consider the the PDE $$A(x, y)\partial_{xx} u + B(x, y)\partial_{xy}u + C(x, y)\partial_{yy}u=h(x, y) $$ Now I want to make a variable substitution $\xi=f(x, y), \eta=g(x,y)$, so I can get $u$ as a ...
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39 views

Simplifying a Vector Integral

While reading the book - Cercignani, Theory and Applications of Boltzmann Transport Equation (I am not a math student), I found this integral which I am unable to understand. Note that $\xi_i , \xi_l$ ...
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1answer
24 views

Calculating a triple integral with spherical coordinates?

I need to calculate $$\iint_D \sqrt{x^2+y^2+z^2} dx dy dz$$ where $D=\{ (x,y,z):x^2+y^2+z^2\leq z\}$ . After substituting $x=r\cos\theta\sin\phi , y=r\sin\theta\sin\phi , z=r\cos\phi $ into the ...
0
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1answer
20 views

Partial differentiation in transformed coordinates

Following lecture notes from MIT it says that, given some variable $A = A(x, y, z(x, y, r, t), t)$ where $r$ is a transformed vertical coordinate $\left. \frac{\partial A}{\partial x} \right|_r = ...
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0answers
27 views

Show that $L[v^2] := \Delta(v^2) + \frac{2}{w} \sum_{i=1}^n w_{x_i} (v^2)_{x_i} \ge 0$

Let $u$ be a function such that $$ \Delta u + \lambda u = 0 $$ for some $\lambda \in \mathbb R$, also let $w$ be a function such that $$ \Delta w + \beta w < 0. $$ for some $\beta \in \mathbb R$. ...
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2answers
24 views

Question on derivation of vector identites and using some symbolic manipulations

Let $f,g : \mathbb R^n \to \mathbb R$, then for the gradient we have the product rule $$ \nabla(fg) = (\nabla f) \cdot g + f \cdot (\nabla g). $$ And by $\Delta(f) = \mbox{div}(\nabla(f)) = \nabla ...