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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13 views

Level sets volume

Suppose that $f:\mathbb{R}^d\to\mathbb{R}$ is a nice function (whatever nice should mean), non-negative, with a compact support. Fix $v >0$ and define $$ A_{\epsilon} := \{x\in \mathbb{R}^d : v \le ...
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1answer
30 views

Inner product, differential forms and surfaces (Stokes' theorem)

I'm trying to understand how do you get the Kelvin-Stokes theorem \begin{equation} \int_{S} (\nabla\times \omega) \cdot \mathrm{d}S = \int_{\partial S} \omega \cdot \mathrm{d}r \end{equation} from the ...
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13 views

Show that the function $f(x,y)=\int_b^yf_2(a,t)\ dt + \int_a^xf_1(t,y)\ dt $ is a potential function

Let $F=(f_1,f_2)$ be conservative over the open rectangle: $$R=\{(x,y):|x-a|<r,|y-b|<r\} $$ I need to show that the function $f(x,y)=\int_b^yf_2(a,t)\ dt + \int_a^xf_1(t,y)\ dt $ is a ...
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2answers
35 views

How to prove a function has no local minima.?

Suppose we have a function $ f:\mathbb{R}^2 \to \mathbb{R}$, of class $C^2$ that satisfies: $3\frac{\partial^2f}{\partial x^2}(x,y)+4\frac{\partial^2f}{\partial y^2}(x,y)=-1$, for all $(x,y) \in ...
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2answers
39 views

Why does this implicit differentiation formula fail?

Suppose we have that $$\frac{dy}{dx} = -\frac{y}{x}.$$ Taking the derivative implicitly with respect to $x$, we can easily obtain $$\frac{d^{2}y}{dx^{2}} = \frac{-\frac{dy}{dx}x + y}{x^{2}} = ...
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2answers
23 views

Do the partial derivatives of this piecewise constant function exist? If yes, how can I compute them?

Given this piecewise constant function $$ f(x,a,b,c,d,e) = \begin{cases} a, & x \lt d; \\ c, & d \le x \lt e; \\ b, & e \le x. \\ \end{cases} $$ do the partial derivatives ...
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42 views

Minimum of $x+y+z$ on $\{(x,y,z) \in \mathbb{R}^3 | z \le x^2+2y^2+3, z\ge 3x+2y\}$

Find the minimum of $x+y+z$ on $$\{(x,y,z) \in \mathbb{R}^3 | z \le x^2+2y^2+3, z\ge 3x+2y\}.$$ My first naive thoughts would be to consider setting up a nasty triple integral and evaluate it or ...
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0answers
16 views

Solving for stationary points for questions of the following type

How do you solve questions like $f(x,y) = x^2y + y^3x -xy$ for stationary points? A link to an educational resource that goes over this would be very helpful as well, as I don't even know what ...
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0answers
18 views

Calculating Hydrodynamic Interaction Tensor

I'm a bit of a newbie when it comes to Tensor calculus. Please excuse me as I learn... Given the Oseen tensor, $\mathbf{T}(\mathbf{R}) = (8\pi \eta R)^{-1} \left[ \mathbf{I} + ...
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1answer
32 views

Geometric interpretation of ${\partial f\over \partial x}= {\partial f \over \partial y}$

I know that $${\partial f\over \partial x}= {\partial f \over \partial y}$$ iff there exists a differentiable function $g$ (of one variable) such that $g(x+y)=f(x,y)$ (where $f : D\subseteq \mathbb ...
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0answers
29 views

Finding maximum value of a 3-variable function using inequality.

Let $a, b, c$ be positive real numbers satisfying $a^2 +b^2+c^2=14$. Find the maximum value of $f(a,b,c)=\frac{4(a+c)}{a^2+3c^2+28}+\frac{4a}{a^2+bc+7}+\frac{5}{(a+b)^2}-\frac{3}{a(b+c)}$
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2answers
36 views

Finding image and inverse function in $\mathbb R^2$

Let $V \subseteq \mathbb R^2$ be open subset which is surrounded by lines: $y=x;~ y=2x;~ x+y=1;~ x+y=3$ and $m:= \frac{y}{x},~ s:=x+y$ for $(x,y) \in V$, so it follows that $y= \frac{ms}{m+1}, ~ x= ...
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2answers
23 views

find the derivative of a function with more than one variable

I have a function $g(a)=f_i(x+a(y-x))$ where a$\in$$\Re$ and x,y$\in \Re^d$. How can I find the first and the second derivative of this function? The second part of the exercise is asking me to use ...
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1answer
59 views

How to partial differentiate a total differential and be rigorous on all the notion?

Start with $$dS=\left(\frac{\partial S}{\partial T}\right)_VdT+\left(\frac{\partial S}{\partial V}\right)_TdV$$ Using the notes shown here Method 1: i) Divide both sides by dV ...
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0answers
18 views

How to derive the required Maxwell relation given that the constant term is not the same?

Question: What is the correct (and mathematically rigorous way) to derive the required Maxwell relation to complete the derivation of "Equation (1.1) becomes" in the text below? (Source: ...
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16 views

For which values of lambda is the set of line integrals bounded above?

Let P = {(x,y,z) $\in$ $R^3$ | 0$\le$ z$\le$1, 1$\le$$x^{2}$+$y^2$$\le$4}. For $\lambda$$\in$R, consider the vector field $$F_\lambda(x,y,z) = (2x+ \lambda y,-\lambda x+2y,2z) $$ in P. For which ...
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1answer
12 views

Finding linear equation for the plane of equidistant point

I'm trying to do some practice questions in my book and I encountered this question: Find a linear equation for the plane consisting of all point $(x,y,z)$ in space that are equidistant from the ...
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2answers
23 views

Evaluate the integral along the stated curve

$\int_{C}{(3x+2y) \, dx + (2x-y) \, dy}$ along the curve y = sin($\pi*x\over2$) from (0,0) to (1,1). (Given that the curve is smooth). Approach: I attempted this problem by parametrizing x = ...
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0answers
28 views

The set composed of domain and codomain of integrable function measure zero

There is this problem which I have constructed a plan to prove, and I am stuck. If anyone could see my plan and tell what is wrong about it I would be very thankful. Let $f: Q \to [0,1]$ be ...
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3answers
29 views

Find $b$ so that $f(x,y) = y^3+3x^2y-15y-12bx$ has some critical point

I am trying to solve this excersice but I can't seem to get to anything but dead ends. Let $b\gt 0$ and $f(x,y) = y^3+3x^2y-15y-12bx$, find all possible values of $b$ so that $f$ has at least one ...
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1answer
32 views

Verify Green's Theorem for region bounded by the lines $x=2$, $y=0$, $y=2x$

Verify Green's Theorem for the region D bounded by the lines $x=2$, $y=0$, $y=2x$ and the functions $f(x,y)=(2x^2)y$, $g(x,y)=2x^3$. I have been trying this question for far too long and I can't ...
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1answer
18 views

Simplifying Double Integrals to Single-Variable Integrals

Let D be a subset of $\mathbb{R}^2$ defined by $ |x| + |y| \leq 1$, and let $f$ be a continuous single-variable function on the interval $[-1,1]$. Show that $$ \iint\limits_D \,f(x+y) \, \mathrm{d}x ...
1
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1answer
47 views

An example of a continuous function on $\mathbb R^2$ with two critical points, both of them minima

Knowing you can not use the minimum bound, there exists a function $f ( x , y )$ continuous in $\mathbb R ^ 2$ that has exactly two critical points which are (both) the minimum? Can you give me an ...
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2answers
24 views

Finding $ \dfrac{\partial z}{\partial x} \text{and}\dfrac{\partial z}{\partial y} $ if $ F(cx - az, cy-bz) = 0 $

If it is given that $ F(cx - az, cy-bz) = 0 $, then find $ \dfrac{\partial z}{\partial x} \text{and}\dfrac{\partial z}{\partial y} $ How do I go about doing this? I don't really understand which ...
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3answers
84 views

Can a non-zero vector field have zero divergence and zero curl?

I don't see how. Curl and divergence are essentially "opposites" - essentially two "orthogonal" concepts. The entire field should be able to be broken into a curl component and a divergence component ...
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0answers
20 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 = ...
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1answer
26 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 ...
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1answer
32 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 ...
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1answer
72 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}|, ...
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1answer
17 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 ...
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3answers
54 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 ...
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38 views
+50

Geometric Interpretation of Antiderivative?

Could someone please give me a geometric interpretation of the above theorem?
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1answer
44 views
+50

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 ...
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22 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) ?
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0answers
13 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 $ ...
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30 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 ...
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2answers
54 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
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2answers
59 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}$ ...
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2answers
32 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 $\Delta u = u_{xx}+u_{yy}$. ...
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1answer
27 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 ...
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1answer
18 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 ...
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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 ...
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3answers
98 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
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1answer
76 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 ...
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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
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1answer
32 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}} ...
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0answers
43 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 ...
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0answers
11 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
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1answer
17 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
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0answers
13 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 ...