Tagged Questions

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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1
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0answers
6 views

Integral on sphere and ellipsoid

Let $a,b,c \in \mathbb{R},$ $\mathbf{A}=\left[\begin{array}{*{20}{c}} \mathbf{a}&{0}&{0}\\ {0}&\mathbf{b}&{0}\\ {0}&{0}&\mathbf{c} \end{array}\right]$ , det A $>1$ Let D = ...
0
votes
1answer
22 views

How to take the triple integral of $ \iiint_G xy\sin (yz)dV$

Hi I'm trying to evaluate $$\iiint_G xy\sin(yz) \ dV$$ where $G$ is the rectangular box defined by the inequalities $0 ≤ x ≤ \pi, 0 ≤ y ≤ 1, 0 ≤ z ≤ \pi/6$. I wasn't sure where to go after the first ...
1
vote
1answer
22 views

Intuition behind Laurent's theorem?

Taylor series has a pretty nice intuitive explanation. If you know the position, velocity, acceleration and so on of a particle you can predict it's location at any time. Does a similar intuitive ...
2
votes
1answer
30 views

$f: \mathbb{R}^2\to \mathbb{R}^2$ is differentiable, and satisfies an inequality that involves its partials - show that f is a bijection.

Suppose that $f: \mathbb{R}^2\to \mathbb{R}^2$ is differentiable, and the partial derivatives of the components $f_1$, $f_2$ satisfy $$max(|\frac{\ df_1}{dx} -1|, |\frac{df_1}{d_y}|, ...
0
votes
1answer
16 views

Prove that both iterated integrals exists but $f$ is not integrable

I need to prove that the function $f$, given by: $$f(x,y)= \begin{cases} 1 \iff (x,y) =(\frac p {2^n},\frac q {2^n}): (p,q,n) \in \Bbb N^3, 0<p,q<2^n \\0 \iff (x,y) \neq (\frac p {2^n},\frac q ...
1
vote
3answers
42 views

Strange double integral

What is wrong with this computation of $\int_0^1\int_{-y}^y \sqrt[3]{x} \, dx \, dy$? I'm considering real functions only. Since $x^{4/3}$ is an antiderivative of the integrand, we will get ...
2
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0answers
16 views

Geometric Interpretation of Antiderivative?

Could someone please give me a geometric interpretation of the above theorem?
1
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0answers
16 views

Geometric interpretation of analyticity?

Suppose the real valued functions $u(x,y)$ and $v(x,y)$ are continuous and have continuous first order partial derivatives in a domain $D$. If $u$ and $v$ satisfy the Cauchy Riemann equations at ...
0
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0answers
14 views

Prove quasiconvexity of a multivariate function [on hold]

I would like to prove that the following function is quasiconvex: $$ f(x_1,x_2) = -x_1*x_2 -1 $$ How do i prove its quasiconvexity (without augmented Hessian matrix) ?
0
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0answers
10 views

Find diffeomorphism from $A=\{y>0\} \cup \{x+y>0\}$ on $B=\{x>0, 0<y<1\}$ and $C=\{x>0, xy>1\}$ to $ D=\{(x,y) : y \ge 0 \}$.

I have got 2 problems about finding diffeomorphisms. Find diffeomorphism from $A=\{y>0\} \cup \{x+y>0\}$ on $B=\{x>0, 0<y<1\}$ Find diffeomorphism from $C=\{x>0, xy>1\}$ to $ ...
3
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0answers
26 views

Implicit Function Theorem exercise

I did an exercise in the book Vector Calculus[Marsden & Tromba] and I check my answer in the book Vector Calculus Study Guide and Solutions Manual[Karen Pao, Frederick Soon] but my answer is not ...
1
vote
2answers
42 views

Show that if $f$ is continuous and non-negative on a compact set $D$ and $\iint\limits_D f\, dA=0$, then $f(a)=0$ for all $a$ in $D$.

I'm trying to show that if $f$ is continuous and non-negative on a compact set $D$ and $\iint\limits_D f\, dA=0$, then $f(a)=0$ for all $a\in D$. My first approach was to argument by contradiction ...
3
votes
2answers
45 views

Definition of a Manifold from Guillimen Pollack

I have been studying differential topology from Guillimen and Pollack (GP). Unlike many other books that define differentiable manifolds using maximal atlases GP starts by saying $ X \subset R^{N}$ ...
0
votes
1answer
13 views

Notation laplace operator squared $\Delta^2$

I have the following expression (in a numerical context) $$\Delta_h u(x) = \Delta u(x) + \frac{h^2}{12} \Delta^2 u(x) + O(h^4)$$ The $\Delta$ is the Laplace operator so $\Delta u = u_{xx}+u_{yy}$. ...
0
votes
1answer
24 views

Determine whether f is a bijection in neighbourhood of singular points

Given a function $f:\mathbb{R}^2\to\mathbb{R}^2$, $f(x,y)=(x^2+y-y^2, 2xy+y)$, determine for which points $(x_0,y_0)$ where $JDf(x_0,y_0)=0$ function $f$ is bijective from some open set containing ...
0
votes
1answer
17 views

Two body problem (rotation around a fixed central point)

Is there a way which isn't physics related, but just using pure maths to find the solution to the following problem: If i have two lines of different lengths at t=0 overlapping each other. They are ...
0
votes
1answer
25 views

Find the value of the triple integral

Find the volume limited by the surfaces $x^2+y^2=4$, $z=0$, together with the portion of the plane $z=x-y$ with $z\ge 0$. So, I've drawn the region of integration and calculated the following ...
1
vote
3answers
96 views

Why is the divergence of $\widehat{r}/r^2$ equal to $0$?

I have read that $\nabla\cdot\dfrac{\widehat{r}}{r^2}$ is equal to $0$. But I cannot understand why. I tried but I cannot solve it. Can anyone explain it please?
3
votes
1answer
68 views

If f ' = 0, then f is constant?

I'm a little confused. After finishing the online multi-variable calculus course from the MIT OCW offerings (I wanted to brush up on the subject more concretely, after my Analysis II course), I ...
0
votes
1answer
22 views

Is there a clean way to derive the gradient of $x^TAx$? i.e. $\nabla_xx^TAx$?

I was trying to take the gradient of $x^TAx$ i.e. $\nabla_xx^TAx$. I did have one idea of how to do this which was expression $x^TAx$ as a double summation and then take the partial derivatives wrt ...
1
vote
1answer
30 views

Convert from Spherical to Cylindrical Coordinates

The following integral is given in Spherical Coordinates, which procedure should I follow to express it in Cylindrical Coordinates? $$\int_{0}^\pi \int_{\frac{\pi}{6}}^{\frac{\pi}{2}} ...
0
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0answers
39 views

basic calculus/analysis question. why does the multivariable chain rule work?

Say $f$ is a function of $x(t)$ and $y(t)$ $$\frac{ df}{dt} = \frac{ \partial f}{\partial x} \frac{ dx}{dt} + \frac{ \partial f}{\partial y} \frac{ dy}{dt}$$ why is it so additively symmetric? (The ...
0
votes
0answers
10 views

Holder continuity and gradient

I am trying to prove the implication of differentiability and constancy from Holder continuity. I have: $\frac{\left\lvert f(x)-f(y) \right\rvert}{x-y} \le M|x-y|^{\lambda} \implies \exists g:x ...
0
votes
1answer
14 views

Subset bounded under linear transformation

Let $T:\mathbb R^n\to\mathbb R^n$ and let $B_r[0]=\{x\in \mathbb R^n : \left \| x \right \|\leq r\}$. Show that $T(B_r[0])$ is bounded. My proof is: $T$ is continuous and $B_r[0]$ is compact (by ...
1
vote
0answers
12 views

triple integral reduction

I have a triple integral of this kind $$\int_0^t{dx f(x)\int_{t-x}^{\infty}{dy g(y)\int_{t-x-y}^{t+\Delta t-x-y}{dz\delta(z)h(x,y,z)}}}$$ where $\delta$ is the Dirac Delta function and the other ...
1
vote
3answers
67 views

Non-existence of $\lim \limits_{(x, y) \to (0,0)} \dfrac{x^3 + y^3}{x - y} $

How to show that $\lim \limits_{(x, y) \to (0,0)} f(x, y)$ does not exist where, $$f(x, y) = \begin{cases} \dfrac{x^3 + y^3}{x - y} \; ; & x \neq y \\ 0 \; \;\;\;\;\;\;\;\;\;\;\; ; ...
4
votes
2answers
67 views

how to compute this definite integral if possible?

how to solve this integral? $$\int_0^a\int_0^a\frac{dx\,dy}{(x^2+y^2+a^2)^\frac{3}{2}}$$ my attempt $$ \int_0^a\int_0^a\frac{dx \, dy}{(x^2+y^2+a^2)^\frac{3}{2}}= ...
0
votes
0answers
18 views

Area, mass and average density of the plane region

The plane region $D$ is bounded by the parabola $y = 2a\left( {1 - {{\left( {\frac{x}{{3a}}} \right)}^2}} \right)$ and the $x$ axis. At the point $(x,y) \in D$, its area density is ${\delta _0} \cdot ...
0
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0answers
14 views

Using Taylor's theorem and Lagrange form of the reminder to prove the second order condition for convexity

I try to prove the second order condition for convexity. So far' I've done the following: First, I prove second order => convexity: Let $f$ be a function with positive semi-definite Hessian. Using ...
0
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1answer
19 views

Find work done by a force field using Green's theorem?

I have to find the work done by a vector field, and while I understand Green's Theorem, I can't set up the limits of integration. The particle moves around the ellipse $x^2+4y^2=4$. How would I set up ...
0
votes
1answer
22 views

Geometric interpretation of duality and Slater's condition

I am trying to study about optimization problems, Lagrange duality and related topics. I came across some presentation on the net, which claims to show the geometric interpretation of the duality and ...
0
votes
0answers
48 views
+50

Finding the Circulation of a Curve in a Solid. (Vector Calculus)

A solid can in spherical coordinates \begin{equation} x=\rho\sin\phi\cos\theta\\ y=\rho\sin\phi\sin\theta\\ z=\rho\cos\phi \end{equation} be described by the following inequalities ...
0
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0answers
11 views

How does cos theta equals to dot product of OP and OQ/ [mod(OP)*mod(OQ)]

O is the origin OP and OQ are vectors, for simplicity lets denote P as vector with magnitude 1 facing angle of 45 degrees from positive x axis. Let's denote Q as vector with magnitude 2 facing ...
2
votes
1answer
39 views

Vector-valued function, proving whether it's continuous, based on its action on any line in R^2:

Suppose $f: R^2 -> R^2$ is a function whose restriction to any line L in $R^2$ is continuous. Prove or find a counterexample: f must be continuous. For starters, I drew an arbitrary point on the ...
0
votes
0answers
23 views

Clarification on to a Solution using Stokes Theorem

The question is there as well, the only part missing is that $F=r$x$(i+j+k)$, where $r=(x,y,z)$. So then $F=(y-z, z-x, x-y)$. The rest is there. I did and understand the problem until after the ...
0
votes
1answer
22 views

Double Integrals and Volume [on hold]

How do I find the volume of a solid in the first octant bounded by the coordinate planes and the graphs of the plane z=3-x-y and the cylinder $x^2+y^2=1$?
3
votes
1answer
74 views

A Deviation from a Conventional Proof of the Basel Problem

There's been many topics on the Riemann-Zeta function, specifically $\zeta(2)$.$$\zeta(2)=\sum_{n=1}^\infty\frac{1}{n^2}=\int_0^1\int_0^1\frac{1}{1-xy}dA$$This is the Basel Problem. Taking the ...
1
vote
0answers
20 views

How to represent real algebraic numbers with period integrals

Background: A real period is defined to be the value of an integral of the form $$\int_D R(x_1,\cdots,x_n)dV$$ where $R$ is a rational function with rational ceofficients, and $D\subseteq\Bbb R^n$ ...
0
votes
0answers
10 views

Expressing integrals of rational functions over domains defined by rational inequalities as differences of volume integrals

Background: A real period is defined to be the value of an integral of the form $$\int_D R(x_1,\cdots,x_n)dV$$ where $R$ is a rational function with rational ceofficients, and $D\subseteq\Bbb R^n$ ...
1
vote
0answers
12 views

Question from Stewart's Calculus regarding proof of independence of path and conservative vector fields.

Please look over this proof. In the proof, it says: "Notice that the first of these integrals does not depend on $x$, so..." How is that so? $C_1$ does depend on $x$. How/Why does it not ...
-5
votes
1answer
31 views

Taylor's theorem in two variables? [on hold]

What would be the expression for the second order Taylor’s formula near a critical point? will it be same as in case of any general point or somewhat different in case we talk about critical points? ...
1
vote
1answer
31 views

Finding the volume of a solid under a region (Triple integrals)

Let S be the solid enclosed above by $x^2+y^2+z^2=2$, below by $x^2+y^2=z^2$ and $y=0$ compute the integral $$ \iiint_S \frac{z}{\sqrt{x^2 +y^2 +z^2} }\text{d}x\,\text{d}y\,\text{d}z $$ What i tried ...
1
vote
0answers
16 views

Finding directional derivatives at $(0,0)$ Multivariate calculus

Given the following function $$f(x,y)=\begin{cases}\frac{x^{707}y}{x^{1414}+y^2}, &(x,y) \neq (0,0)\\0 ,&(x,y)=(0,0) \end{cases}$$ Does $f$ have a directional derivative in every direction ...
1
vote
1answer
26 views

Finding critical points of a multivariable function

Let $f(x,y)=e^{x^2-xy+y^2}$ (a) Find all the critical points of the following function. (b) Find the all the local maxima and local minima of the function if there is any. What i tried. I tried ...
2
votes
2answers
15 views

Showing that normal line passes through a point.

I need to show that a line passes through a point. How should I go about doing this? The question is: Let $L$ be the normal line at $(1,1,1)$ to the level surface of $f(x,y,z) = x^2 - z$ that ...
0
votes
1answer
17 views

Finding Absolute Min/Max with given Domain and Equation. f(x,y)

Question is: Suppose that $f(x,y) = 5x+3y$ at which $-3 \leq x$, $y \leq 3$. Find Absolute minimum and maximum of $f(x,y)$. Since $\frac{\partial}{\partial x} f(x,y) = 0$ or ...
0
votes
0answers
6 views

Needs clarification about statements related to continuity , differentiability (Multivariable)

I have recently done limits,continuity of multi variable functions ,but i feel i need clarification as to which is true or not regarding these statements . Which of these are true ? A . If partial ...
0
votes
0answers
30 views

Solving the PDE $\frac{\partial u}{\partial t}=a\frac{\partial^2 u}{\partial x^2}+b\frac{\partial u}{\partial x}$

I am trying to solve the PDE $\frac{\partial u}{\partial t}=a\frac{\partial^2 u}{\partial x^2}+b\frac{\partial u}{\partial x}$ for constants $a$ and $b$ with conditions $\frac{\partial u}{\partial ...
0
votes
1answer
13 views

Finding derivative at specific time on space curve

I am trying to do some practice questions in my book, but I don't know how to do this specific question: Suppose the function $F(x,y,z,t)$ satisfies $F_x(3,9,18,3)=1$, $F_y(3,9,18,3)=-2$, ...
5
votes
1answer
42 views

One Question about the Fubini's Theorem

The Fubini's Theorem says: If function $f:X \times Y \rightarrow R$ is integrable over $X \times Y$, then $$ \int_{X \times Y}f(x,y)dxdy = \int_{X}dx\int_{Y}f(x,y)dy = \int_{Y}dy\int_{X}f(x,y)dx. $$ ...