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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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7 views

Weak Solutions Diffusion Equation

Let $\Omega$ be a bounded open set of $\mathbb{R}^n$, with smooth boundary $\Gamma$. Consider an evolution equation in the cylinder $Q=\Omega\times ]0,\,T[\quad T>0, \text{finite}$. Set $\sum=\...
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10 views

How to know whether to choose the x-bound or the y-bound for this triple integral

In my textbook for calculus 3, I have been working on example of the triple integral. Though I do know polar, cylindrical, spherical coordinates, this section of the book expects you to work with ...
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16 views

Change of variables in triple integral

Let $D$ be the region in $xyz-$space defined by inequalities $1 \le x \le 2, 0 \le xy \le 2 $ and $0\le z \le 1$. I want to evaluate $\displaystyle \int\int\int_D (x^2y + 3xyz) \text{dxdydz}$ by ...
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1answer
22 views

Transform coordinate system for the gradient of a function at a specific x, y, z value.

I have the gradient of a function $f(x,y,z)$ at a specific value of $x, y,$ and $z$ in the vector form: $$ \nabla f(x,y,z)\Bigr|_{\substack{x=x_1\\y=y_1\\z=z_1}} =\begin{pmatrix} \frac{\partial{f}}{\...
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14 views

A vector calculus formula

Let $A, B$ be vector fields in $\mathbb R^3$. We have $$ \text{curl}\bigl((A\cdot \nabla)B\bigr)=(A\cdot \nabla)\text{curl}B -((\text{curl}A)\cdot \nabla)B+R(A,B). $$ I know that $R(A,A)=0$ and I ...
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1answer
16 views

One to one relation between functions that coincide in certain values

I was reading a proof of multivariable calculus that suggested the following property might be true: Given two functions $f(x)$ and $g(x)$ in $\mathbb{R}^n$, if $\forall x_1,x_2$ $f(x_1)=f(x_2)$ $\...
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2answers
45 views

Finding the global minimum of $\int_{0}^{1} \left( ax+b+\frac{1}{1+x^{2}} \right)^{2}\,dx$ having just the local minimum.

In order to calculate the values of $a$ and $b$ such we get the minimum possible for: $$\int_{0}^{1} \left( ax+b+\frac{1}{1+x^{2}} \right)^{2}\,dx$$ I got the help of @TheSimpliFire among others to ...
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3answers
38 views

Find distribution of $Z=\frac{X+Y}{2}$ given $f_{X,Y}(x,y)=e^{-(x+y)}$

Excercise Let $X, Y$ be random variables such that their joint density function is defined by: $f_{X,Y}(x,y)=e^{-(x+y)}, \enspace x,y>0$. Find the distribution of Z defined as: $Z=\frac{X+Y}{2}...
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1answer
22 views

Is Schwarz's Theorem valid for higher order derivatives?

I know that $\frac {\partial^2 f}{\partial x \partial y} = \frac {\partial^2 f}{\partial y \partial x}$ when $\frac {\partial f}{\partial x}$, $\frac {\partial f}{\partial y}$ exist and are continuous,...
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2answers
168 views

How to compute a Jacobian using polar coordinates?

Consider the transformation $F$ of $\mathbb R^2\setminus\{(0,0)\}$ onto itself defined as $$ F(x, y):=\left( \frac{x}{x^2+y^2}, \frac{y}{x^2+y^2}\right).$$ Its Jacobian matrix is $$\tag{1} \begin{...
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How to convert spherical co-ordinates of a vector field to cartesian co-ordinates :

v$(r,φ,θ) = (r cos2 θ)$r$ − (rcosθsinθ)$θ$ + 3r$φ, where r, θ and φ are the unit spherical vectors. I was trying to calculate the line integral of the function along the path described in the ...
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1answer
28 views

Existence of a function with $||grad f||>\epsilon$

I want to construct a function $f$ on the unit ball $B$ of $\mathbb{R}^n$, such that it is negative on a closed subset of the boundary $\partial'B\subsetneqq\partial B$, zero on a given point $p\in B$,...
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1answer
32 views

Understanding the domain of the triple integral for $f(x,y,z)=x+2y+z^2$

So, I am having trouble (again) with the domain for a triple integral of a function, bounded by the paraboloid $2y^2=x$ and the $x+2y+z=4$ and $z=0$ planes I have tried to guess the bounds for x,y ...
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2answers
19 views

Application of the Divergence Theorem with change of variable

Let $S$ be the ellipse $\left(\frac{x}{a}\right)^2+\left(\frac{y}{b}\right)^2 + \left(\frac{z}{c}\right)^2=1,$ with $\vec{n}$ oriented outwards. Compute $\int\!\!\!\int_S \vec{F}\cdot \vec{n}\,dA$ for ...
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0answers
13 views

Partial differential Equation uniqueness

Let $\Omega\in\mathbb{R}^{n}$ be a bounded connected open set. I have the following partial differential Equation; \begin{align} \nabla\cdot\left(-D(x)\nabla \psi\right)&=F\quad \text{in}\quad \...
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0answers
13 views

Reference for the multivariate Leibniz rule of many factors

I'm looking for a reference (a book/article) with a formula to $$ \frac{ \partial ^ k }{ \partial x_1^{k_1} ... \partial x_n^{k_n} } f_1(x) ... f_m(x) , $$ where $k=k_1+...+k_n$, $x=(...
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0answers
30 views

If $f(x,y)=9-x^2-y^2$ if $x^2+y^2\leq9$ and $f(x,y)=0$ if $x^2+y^2>9$ study what happens at $(3,0)$

If$$f(x,y)=\begin{cases}9-x^2-y^2&\text{if }x^2+y^2\leq9\\0&\text{if }x^2+y^2>9\end{cases}$$study the continuity and existence of partial derivative with respect to $y$ at point $(3,0)$. ...
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1answer
57 views

Partial Derivative Disambiguation

There are at least two substantially different meanings to $\frac{\partial}{\partial x}f(x,\ y,\ z(x))$. The $\partial x$ could mean "with respect to $x$ the independent variable," or it could mean "...
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1answer
25 views

ODE in $\mathbb{R}^n$ defined by the gradient of a function

I'm studying for an exam and I got stuck in this question: Let $x: I \to \mathbb{R}^n$ be a differentiable parametrized curve (I is an interval) in $\mathbb{R}^n$ and $f: \mathbb{R}^n \to \mathbb{R}$...
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0answers
25 views

Chain rule for a transformation from $\mathbb{C}$ or $\mathbb{C}^2$ to $\mathbb{R}^2$

Note that, for $z=x+iy\in \mathbb{C},$ $x=\frac{z+ \bar z}{2}\in \mathbb{R}~\hbox{and}~y=\frac{z-\bar z}{2i}=-\frac{i}{2}(z-\bar z)\in \mathbb{R}.$ I was wondering if the following is true: $$ \frac{\...
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1answer
43 views

Computing partial derivatives of $f(a,b)= \int_{0}^{1}(ax+b+\frac{1}{1+x^2})^{2}dx$ using chain rule.

Let $f(a,b)= \int_{0}^{1}(ax+b+\frac{1}{1+x^2})^{2}dx$. I want to compute $\frac{\partial{f(a,b)}}{\partial{a}}$ and $\frac{\partial{f}(a,b)}{\partial{b}}$. I was told in the text that $$\frac{\...
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1answer
19 views

Finding the curve of intersection of a cylinder and cone

I have a cone $x^2 + y^2 -z^2 =0$ and a cylinder $ x^2 +y^2 -2ax =0$. Together they look like this. If one were to project the intersection onto the xy plane, the curve given by this intersection ...
3
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2answers
21 views

Showing a mapping is bijective if and only if a matrix is invertible

Let $\mathbf{A}$ be an $n\times n$ matrix and let $\mathbf{c}$ and $x_{\star}$ be point in $\mathbb{R}^{n}$. Define the affine mapping $\mathbf{G} : \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ by ...
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1answer
24 views

Understanding the domain of the triple integral for $f(x,y,z)=x^2+y^2$

So, I am having trouble with the domain for the triple integral of $f(x,y,z)=x^2+y^2$, bounded by the paraboloid $x^2+y^2=2z$ and the $z=4$ plane I am currently trying to project it on the XY axis, ...
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1answer
20 views

Checking saddle point or not - using rules of 'Fundamental Theorem of Calculus'

True or false For the function, $f(x,y)=\int_{2x}^{-y+2{x^2}}e^{-{t^2}}dt$ $(0,0)$ points is the saddle point. I can do it the long way by solving the integral first but I believe there is a way to ...
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0answers
16 views

Proving $\arg$ restricted to an open subset of $S^1$ is smooth

Let $U$ be an open subset of $S^1\subset \mathbb{R}^2$. Define $\theta:U\to \mathbb{R}$ by $\theta(x,y)=\arg(x+iy)$ where $\arg$ is the principal argument. I want to prove that $\theta$ is smooth (i.e....
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0answers
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Can we perturb a map to have distinct singular values?

Let $\mathbb{D}^n$ be the closed $n$-dimensional unit ball, and let $f:\mathbb{D}^n \to \text{GL}_n^{-}$ be smooth. ($\text{GL}_n^{-}$ is the set of $n \times n$ real matrices with negative ...
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4answers
62 views

What's “limit doesn't exist” multiplied by limit that equals zero?

OK, we're having a strong discussion just a day before our Calculus exam. The problem's next: To check if this function is continuous: $$\frac{y^2\,\sin x}{x^2 + y^2}$$ at (0,0). We get DNE $\cdot 0$, ...
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1answer
44 views

Find the area between $r=a\cos(\theta)$ and $r=a(1+\cos(\theta))$

So, I have to calculate an integral with a domain limited by two functions: $r=a\cos(\theta)$ and $r=a(1+\cos(\theta))$ , where $a>0$ The issue here is that I cannot wrap my head around what the ...
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1answer
16 views

How does Green's theorem imply the divergence theorem in the plane?

Both Green's theorem and Stokes' theorem involve the integral of a curl and it is easy to see that Green's theorem is a planar version of Stokes' theorem. However, the divergence theorem involves the ...
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22 views

Prove Jacobian of $f: \mathbb{R}^{2} \to \mathbb{R}^{2}$ with 3 conditionals over $\mathbb{R}^{2}$ is $I_{2 \times 2}$.

If $f: \mathbb{R}^{2} \to \mathbb{R}^{2}$ is given by: $f(x,y)= \begin{cases} (x,y-x^{2}) & if & x^{2} \leq y \\ (x,\frac{y^{2}-x^{2}}{x^{2}}) & if & 0 \leq y \leq x^{...
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1answer
31 views

Picture flow of ODE

Consider the ODE $$\begin{cases} \frac{d}{dt}\Phi(x,t) = f(\Phi(x,t),t) \quad t >0, \quad x \in \mathbb R^2 \\ \Phi(x,0) = x, \quad x \in \mathbb{R}^2 \end{cases}$$ Suppose that the flow $\Phi$ ...
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0answers
15 views

Estimating multivariate random walk plus noise model using OLS

I'm currently working on replicating a scientific paper for practice in which they estimate a multivariate random walk plus noise model, apparently using OLS. I have no clue however how they would do ...
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0answers
7 views

Compute the flux of $F(x,y,z)=(2x,2y,2z)$ through $S=\{(x,y):x^2+y^2=9,0\leq z\leq 5\}$

Compute the flux of $F(x,y,z)=(2x,2y,2z)$ through $S=\{(x,y):x^2+y^2=9,0\leq z\leq 5\}$ So I parametrized $S$ by $\sigma(r,\theta)=(r\cos \theta,r\sin \theta,r^2)$ where $0\leq r\leq \sqrt{5}$ and $0\...
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0answers
14 views

Will lagrange multiplier method find all stationary points, or just minima and maxima?

Let $f,g : \mathbb{R}^n \to \mathbb{R}$ be smooth functions and $U_c = g^{-1}(c)$ for each $c \in \mathbb{R}$. For each $\lambda \in \mathbb{R}\setminus\{0\}$, consider the equations $$ \nabla g = 0 \...
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1answer
19 views

Why do we take magnitude into account when calculating the directional derivative?

Given that the directional derivative is defined formally as: $$ \nabla_\vec{v}\, f\left(\vec{x}\right) = \lim_{h \to 0} \frac{f\left(\vec{x} + h\vec{v}\right) - f\left(\vec{x}\right)}{h|\vec{v}|} $$ ...
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0answers
29 views

Triple integral to calculate the volume of pyramid vs it's formula

It is well known the formula to calculate the volume of a pyramid: $V=\frac {1} {3} bh$, where where $b$ is the area of the base and $h$ the height from the base to the apex. However I need to ...
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1answer
28 views

Question about the gradient of a composite function

I am new to calculus and am trying to work out the following question, with no success so far… Any feedback would be great! Within function $f(x,y)$, variable $y$ is a function of $(x,z)$, in other ...
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1answer
21 views

Question about gradient in a complex function

I am new to calculus and cannot see the logic of the following question… Any feedback will be really appreciated! The function $f(x,y,z)$ is differentiable at all points, and satisfies $f(x,y,2x^2+y^...
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2answers
23 views

Expand the given function in an appropriate cosine or sine series. (Fourier series)

$$f(x) = \begin{cases} x-1, \quad& -\pi < x <0 \\ x+1, & 0 \leq x \leq \pi \end{cases}$$ This is odd so we expand with an odd series $$b_n = \frac{2}{\pi} \int_{0}^{\pi} (x+1) \sin(nx) ...
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1answer
36 views

Fourier series confusion

Find the Fourier series of Question 1 $$ f(x) =\begin{cases} 0&&\text{for $-1 < x < 0$}\\\\ x&&\text{for $0 \leq x \leq 1$} \end{cases} $$ and Question 2 $$f(x) = x + \pi,\;...
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2answers
27 views

Prove the vector calculus identity $\frac{1}{2}\mathbf{\nabla(\lVert u \rVert ^2) = (u \cdot \nabla)u+u \times (\nabla \times u )}$

My attempt: Consider the first component of both sides. $$LHS=\frac{1}{2}\frac{\partial}{\partial x_1}(u_1^2+u_2^2+u_3^2)=u_1 \frac{\partial u_1}{\partial x_1}+u_2 \frac{\partial u_2}{\partial x_1} + ...
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1answer
38 views

Selling inventory while maximizing the present value of profit

A company owns an inventory of $100$ units of a good. It must sell the entire inventory over the next three periods. The profit function for sales within any one period is $$\pi(x_t) = 50x_t − ...
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1answer
25 views

Showing that $f: D_1(0) \subset \mathbb{R}^2 \to \mathbb{R}, f(x) = \frac{1}{1- \|x\| }$ is continuous

I understand that I need to show that for all $x_0 \in D_1(0)$, $$\lim_{x\to x_0} \frac{1}{1- \|x\| } = \frac{1}{1- \|x_0\| }$$ But I have trouble bounding the denominator, $$\bigg| \frac{1}{1- \|x\...
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1answer
37 views

Integration on Differential Forms.

So I know that divergence and curl of a vector field $F$ can be related to a differential form $\alpha$ by div $ F = \star d \star \alpha$ and curl $ \cdot F = \star d \alpha $ , where $\star$ is the ...
11
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2answers
173 views

Find $a$ and $b$ for which $\int_{0}^{1}( ax+b+\frac{1}{1+x^{2}} )^{2}\,dx$ takes its minimum possible value.

Calculate for which values $a$ and $b$ the integral $$\int_{0}^{1} \left( ax+b+\frac{1}{1+x^{2}} \right)^{2}\,dx$$ takes its minimum possible value? For being honest I'm not sure how to ...
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0answers
25 views

Why can't these two mappings be bijective?

Let $\phi : \mathbb{R}^{2} \rightarrow \mathbb{R}$ be a continuously differentiable function and define the mapping $\mathbf{F} : \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ by $$\mathbf{F}(x, y) = (\...
1
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0answers
42 views

Total differential and overestimation

Consider the function $f(x,y)=10-(x^2+y^2)$ at (1,1) with dx=2 and dy=3 The function value at (1,1) is 8. Computing the change in the function by taking a first order total differential gives us $-(...
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1answer
22 views

Double integral to calculate using Polar coordinate system?

I've met a double integral which seems to be calcultated in polar system. $\iint_D \frac {y} {x^2+y^2} dxdy$ , where $D$ is the region bounded by the following conditions: $-2y \le x^2+y^2 \le -4y$; ...
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1answer
24 views

Why is the normal vector different in cartesian coordinates vs. spherical coordinates?

Consider the sphere $x^2+y^2+z^2=1$. Let $\mathbf x(u,v)$ be a parameterization for the sphere. Say I was trying to find specifically the normal vector given by $$ \frac{\partial \bf x}{\partial u} \...