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Questions tagged [multivariable-calculus]

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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System of 4 equations with 4 unknowns in Excel: stress and strain evolution during temperature cycles (hysteresis loop)

I am trying to solve the following system of $4$ equations with $4$ unknowns (in red): $$\begin{cases} \color{red}{\gamma_{iv}} + \frac{\color{red}{\tau_{iv}}}{1372} = 4.24 \times 10^{-4} (T_{i + 1} -...
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Decompose a function of multiple variables to product of functions of single variable

For simplicity, we consider a two variable function $f = f(x,y): \mathbb{R}^2 \to \mathbb{R}$. As stated in this post, in general, we can't write $$f(x,y) = g(x) h(y)$$ as a product of single variable ...
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Converting Integral to Spherical Coordinates with Unit Sphere

Can someone please explain how to convert this integral to spherical coordinates? Here, $S^{n-1}$ is the unit sphere in $\mathbb{R}^n$ and $y=x+tw\in$ $B(x, 2r)$ $\subset \mathbb{R}^n$ where $0<t&...
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Integrate $\frac1{\sqrt{u^2+v^2} \left(1+u^2+v^2\right)}$ over $[-1,1]^2$?

Problem $$ \text{Evaluate} \quad I = \iint\limits_{[-1,1]^2} \frac{du\,dv}{\sqrt{u^2+v^2} \left(1+u^2+v^2\right)}$$ The integral is a special case of one that appeared in this recent post, in which ...
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Why is this bound correct?

I recently encountered the following statement: Let $S:=\big\{(x,y,z)\in\mathbb{R}^3 \mid x>0, y>0, z>0, x+y+z<7\big\}$. Let $f : S \to \mathbb{R}$ such that: $$f(x,y,z) = \ln x+2\ln y +3 \...
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Notion of inverse for maximum of linear functions

Assume we have $J\in\mathbb{N}$ linear pieces of the type $a_{j}^Tx+b_{j}$ with $a_j\in\mathbb{R}^n$ and $b_j\in\mathbb{R}$ for all $j\in [J]$. Consider the function $f(x) = \max_{j\in[J]}a_{j}^Tx+b_{...
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composite derivative and variable change

I'm asking for help with this calculation that I've been racking my brain over for two days! We have a variable change: $$p=e^{-\gamma t}\hat p$$ $$dp = d \hat p e^{-\gamma t}$$ Now the pdf becomes $$\...
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Changing coordinates of $2$nd order partial operators

Let's work in $\mathbf R^n$. If we want to change coordinates $\mathbf x\to\mathbf r$, with them related like $$ \mathbf x=\mathbf x(\mathbf r) $$ Then, the second order generic partial operator in ...
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Regular exhaustion of a bounded domain $\Omega$ with smooth boundary.

A bounded domain $\Omega\subseteq\mathbb{R}^n$ with smooth boundary (Here smooth means $\partial\Omega$ is a (n-1) dimensional manifold) is said to have regular exhaustion iff 1)There exists a ...
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How to integrate over this multi-function surface?

I am trying to get the average height of a surface composed of multiple functions, but my limited understanding of Calculus is keeping me from doing so. I am interested in integrating over the ...
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Compacting Notation of Propagation of Uncertainty Formula into Vectors with Covariance?

This question has a nice solution to compacting the propagation of uncertainty formula when all terms are uncorrelated. To re-iterate, they found that the combined uncertainty, which I will denote $...
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How do I evaluate $\int_0^1 \int_0^\infty \frac{\ln t}{1 + \alpha t^2} \mathrm{d}t \, \mathrm{d}\alpha$? [closed]

How would I evaluate the integral $$ \int_0^1 \int_0^\infty \frac{\ln t}{1 + \alpha t^2}\, \mathrm{d}t \, \mathrm{d}\alpha? $$
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Why is $\int_0^1{df(tx_1,\dots, tx_n)\over dt}=\int_0^1\sum_{i=1}^n{\partial f\over\partial x_i}(tx_1,\dots,tx_n)\cdot x_i\;dt$?

For context: Milnor's Morse theory page 6, Lemma 2.1: $V$ is a convex neighborhood of $0$ in $\mathbb R^n$. also, $f,g \in C^\infty \left( V \rightarrow \mathbb R^n\right)$ where $f(0) = 0$ and $...
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Laplacian of arbitrary power of arbitrary norm

So I have the function $f : \mathbb R^d \to \mathbb R$ given by $f(x) = \lVert x \rVert_p^q$, where $\lVert \cdot \rVert_p$ denotes the $p$-norm on $\mathbb R^d$, given by $$ \lVert x \rVert_p = \left(...
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Finding all $ f: \mathbb{R}^2 \to \mathbb{R} $ s.t.: $ \frac{\partial f}{\partial x} - 3 \frac{\partial f}{\partial y} = 0 $

I need to find all continuously differentiable functions $ f: \mathbb{R}^2 \to \mathbb{R} $ such that $ \frac{\partial f}{\partial x} - 3 \frac{\partial f}{\partial y} = 0. $ I was given the hint ...
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How to deduce the general case of Green's theorem from rectangular case?

The proof of Green's Theorem for rectangle is very simple. The proof for triangle is not too bad, and the general case follows ( at least intuitively ) from the triangle case. But I'm lazy. So is ...
new account's user avatar
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Finding function for gradient field

I am probably just stuck with some stupidity in my brain but I cannot reason what is going wrong in this problem. I am doing multivariable calculus MIT course and I am also doing exercises from ...
exbluesbreaker's user avatar
1 vote
2 answers
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Implicit differentiation choice

I was reading Calculus early transcendentals by Howard Anton, in which I encountered an example as follows, Find the slope of tangents of a sphere $x^2+y^2+z^2=1$ in the direction of $y$ at points $(2/...
Kaustubh Limaye's user avatar
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Approximating the double integral of cosh(P) over [0,1]×[0,1] as mean(cosh(P))

On this website: https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.root.html Towards the bottom of the URL, under the heading of "Large Problem", a python code is given to ...
GeorgeO's user avatar
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Rewrite linear expression in second derivative as divergence of function of first derivatives [closed]

consider a function $f(x)$ of two variables $x^i$, $i =1,2$, and the expression $F = [A^{ij}(\partial f)] \, \partial_i \partial_j f$, where $A^{ij}$ depends on the first derivatives of $f$, i.e., $\...
James's user avatar
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Sketchy use of multivariable chain rule under too weak hypoteses

I've found this statement in my real analysis course notes: Let $f: B_r (x_0) \subseteq \mathbb{R}^m \to \mathbb{R}$ ($B_r (x_0) = \{ x \in \mathbb{R}^m : d(x, x_0) < r \}$) be a function such ...
Rick Does Math's user avatar
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1 answer
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Can the partial derivative of a multivariable function be such that it does not depend on the other variable? [duplicate]

I wanted to know if there is a multivariable function out there which when partially differentiated with respect to x returns a function only in x.(Not dependent on y(the other variable)). For example-...
Aditya Saraswat's user avatar
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1 answer
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Help with multivariate function derivative algebra... Middle terms don't match as expected

Intro I want to compute $\frac{\partial^2 f}{\partial m^2}$ where we define $ f(m,\sigma) = g(m, I(m, t)) $ I asked about this in a prior question, however I'm realizing that I don't actually ...
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Question about Lemma 19.1 in Munkres' Analysis on Manifolds

In Munkres' Analysis on Manifolds, page 162 Lemma 19.1 Step 2 it states: Third, we check the local finiteness condition. Let $\mathbf{x}$ be a point of $A$. The point $\mathbf{y}=g(\mathbf{x})$ has a ...
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I don't understand the notation in Mechanical engineering. Could you help me to understand it better?

Consider a particle $M$ occupying the position $\vec{a}$ at time $t = 0$ and the position $\vec{x}$ at time $t$. A function $f = f(M, t)$ associated with a particle $M$ (e.g., its velocity, ...
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Question on Continuum Mechanics: How does Eulerian derivatives of $f$ and Lagrangian derivatives of $f$ different?

Consider a particle $M$ occupying the position $\vec{a}$ at time $t = 0$ and the position $\vec{x}$ at time $t$. A function $f = f(M, t)$ associated with a particle $M$ (e.g., its velocity, ...
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1 answer
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Taking the derivative of a generic function... Not sure how to get a factor of 2

I'm working on a problem involving a function and its nested dependency. The problem statement provides a PDE and its derivatives, and I'm trying to understand how the second derivative term is ...
financial_physician's user avatar
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2 answers
66 views

Integration by parts on an area

I'm reading an Engineering book. All I can think of is integration by parts $$\int_{\Omega} \dfrac{\partial M_y}{\partial x}v\text{d}\Omega = M_y v|_{?}^{?} - \int_{\Omega} M_y\dfrac{\partial v}{\...
user900476's user avatar
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1 answer
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Local minima of a multivariable function

Let $f : \mathbb{R}^{2}\setminus \{(0,0)\} \to \mathbb{R}$ be the function determined by $f(x,y) := (x^2+y^2) log(x^2+y^2)$. It is not hard to prove that all points in $C:= \{(x,y)\in \mathbb{R}^{2} : ...
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1 answer
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Triple integral with absolute values in bounds and function [closed]

\begin{align*} \mbox{Let}\ f(x_1,x_2,x_3) & = {\rm e}^{-2\vert x_3 \vert}\ \\[3mm] \mbox{and}\ A & = \left\{\left(x_1,x_2,x_3\right)\in\Bbb R^3\ \mid\ x_1^2 + 4x_2^2 \le \left\vert x_3 \right\...
Michele Lausdei's user avatar
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Prove that $h'(t)=\ell'(t).$ How does this proof shows The vector $U$ is independent of the reference time $t_0?$

The velocity of the material point $M$ occupying the position $\vec{x}$ at time $t$ is the vector $\vec U=\vec U(\vec x, t)=\frac{\partial \Phi(\vec a,t,t_0)}{\partial t}.$ Where $\vec x =\Phi(\vec a,...
Unknown x's user avatar
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2 votes
3 answers
126 views

Minimum value of $|z^4+z+\frac{1}{2}|$ on the unit circle

Let $z$ be a complex number. What is the minimum value of the expression $|z^4+z+\frac{1}{2}|$ for $|z|=1$? I wanted to explore the long process of considering $z=x+iy$, and substituting to get the ...
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Questions regarding $C_n^1(\overline\Omega)$, the space of functions with normal derivatives

The definition of which functions have normal derivatives, and to which we can apply Green's First Identity to, seems to be very delicate. Let $\Omega$ be a $C^1$ domain in $\mathbb{R}^d$ with $d\geq ...
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What's wrong with my counter-example to an exercise in Tapp's Differential Geometry of Curves and Surfaces?

Exercise 1.23 in Tapp's Differential Geometry of Curves and Surfaces asks: "Let $\gamma : I \to \mathbb R^3$ be a regular space curve, and let $P \subset \mathbb R^3$ be a plane that does not ...
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Multivariate Chain Rule with Derivatives in Intermediate Functions

I have a function $$G: \mathbb R^d \times \mathbb R \times \mathbb R^d \to \mathbb R$$ where $d$ is a positive integer and the arguments of $G$ are denoted by $({\bf y}, z, {\bf p})$. I'm denoting ...
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Using the Chain Rule to Switch Between Coordinate Axes in a PDE

I have a system of PDEs that are written in terms of two coordinates: $(x,z)$ which are the usual Cartesian coordinates and $(l,n)$ which are tangential and normal components to some deformable ...
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How to evaluate the exact value of the sum $\sum_{x\in \Lambda_N}\frac{1}{\|x\|_2^3}$ in 2d?

Let $\Lambda_N=\{1,2,\dots,N-1,N\}^2\subseteq\mathbb{Z}^2$ be a set of lattice points in 2d. Could we evaluate the exact value of the sum below? $$(*)\quad\sum_{x\in \Lambda_N}\frac{1}{\|x\|_2^3},\...
Chang's user avatar
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Struggling to reconcile definition of surface integral with physical situations

I have a confusion about how only including the component of a vector field normal to a surface in surface integrals gives the intuitively correct answer in certain physical situations. I saw in an ...
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2 answers
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Stokes’ theorem question for cylinder-like surface with two disjoint boundaries

Q: Consider $S=\{(x,y,z)\mid x^2+y^2+z^2=25,\; 0≤z≤4\}$ and the vector field $\vec F(x,y,z)=(y,-z,x)$. Use Stoke’s theorem to calculate $\iint_S(\nabla \times \vec F )\cdot \vec n \; dS$ $\vec n$ is ...
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Introductory questions to fourier transform: help with the visualization of integration (with respect to one axis) of a 2D function

I'm not very familiar with more advanced concepts, but I have a decent understanding of single variable calculus with real-valued functions $f: \mathbb R \to \mathbb R$ (via Spivak). I wanted to learn ...
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How to prove that the area of a continuous function $ f\left( x \right) $ enclosing a curved trapezium with the x-axis can be n-equalised

First is a simple problem for a continuous function $f(x)$ defined on the interval $[0,1]$ with $f(x)>0$. Prove that: on the interval $[0,1]$, there exists $x=x_0$ such that $x=x_0$ bisects the ...
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Inverse higher order derivatives of a function with respect to changes in it's own gradient

I have an optimization problem operating on the implicit function $$F(x, y, z) = 0$$ and a vector $$(u, v, w)$$ where the solution to the problem is the point $(x_0, y_0, z_0)$ that satisfies the ...
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0 answers
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Least Squares Function Approximation and Convexity of Functions

I have been reading about Least Squares function approximation and am dealing with the following definition: Let $f$ be continuous on $[a,b]$ and let $W$ be a finite dimensional subspace of $C[a,b]$. ...
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3 votes
3 answers
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Need help with evaluating a multivariable limit

Tried evaluating the following limit using the squeeze theorem: $$\lim_{(𝑥,𝑦)→(0,0)} \frac{1-\cos(xy)}{xy^2}$$ $$0 \leq f(x,y) \leq \left| \frac{1-\cos(xy)}{xy^2} \right| \leq \left| \frac{1-\cos(...
tachy's user avatar
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1 vote
1 answer
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Using difference quotients to find the surface area of a parametric surface

I am reading Stewart’s book on multivariable calculus to brush up before reading about electrodynamics and I encountered the following attempt to define the surface area of a parametric surface, which ...
Joa's user avatar
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Multivariable taylor - why does $R_k(a) = o(||h||^k)$

Let $f : E \to \mathbb{R}$ be $C^k$ for $E \subset \mathbb{R}^n$ open. Choose $a \in E$ and $h \in \mathbb{R}^n$ such that $a + h \in E$. Then $\exists \theta \in (0,1)$ such that $f(a + h) = f(a) + (...
Camiel's user avatar
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1 answer
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This function has no saddle points: correctness of this reasoning

I would like to know if my reasoning is correct. I have the function $f(x, y) = e^{3x}(1+25x^2+25y^2)$ and I have to study the stationary points. After computing the gradient I found $$\begin{cases} 3 ...
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1 answer
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how to generalize a version of IVT theorem

We know, with the IVT theorem: if $f(0)=0,f'(0)>0$ and $f(a)<0$ for a continuous function, one can deduce that there exists at least $c \in ]0, a[$ such that $f(c)=0$. could we generalize this ...
Sayed Sayari's user avatar
5 votes
1 answer
137 views

Integral calculated directly and with Gauss Green formula

I checked on Wolfram Alpha that the directly calculated integral was correct, when I go to use the Gauss Green formula, I get $0$ as a result, when instead, I ...
Pizza's user avatar
  • 179
2 votes
2 answers
73 views

Polar coordinates not to prove that the limit goes to zero

I have to study the differentiability in $(0,0)$ of the function $f(x,y)=\frac{x|{y^k}|}{\sqrt{(x^2+y^2)^3}}$ for $(x,y)\neq(0,0)$ with $k\geq0$, and $f(0,0)=0$. By definition of differentiability, ...
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