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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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5 views

Volume of a defined region using triple integrals

I was doing some excercises and I came upon this one, but I couln`t define the limits of integration. The problem says the following: "Find the volume of the region defined by: z = x^2 + 3y^2 ; z = ...
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Integral and norm

Consider the fonction : $N(x,y)=\int_{0}^{1}\mid x+ty\mid dt $ where $x,y$ are real numbers. I have to proove that this function defines a norm $R^2$. First of all, I said that, by linearity : $$N(\...
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3answers
25 views

Prove that the second order implicit multivariable derivative equals $0$

Let $ y $ be determinaded implicitly as a function of $ x $ by the equation $ x^2 + y^2 + 2axy = $ 0, with $ a > 1 $. Prove that $ \frac{d^2y}{dx^2} = 0 $. My failed attempt $$ \frac{dy}{dx} = - \...
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1answer
17 views

How can I know if the partial derivatives exist in this case?

Consider the function: f(x,y) = { abs(x)*y/sqrt(x^2+y^2) when (x,0) != (0,0) 0 when (x,0) = (0,0) } How can I know if the partial derivatives exist and how do i find them? In addition, can this ...
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26 views

Total derivative in polar coordinates

Let's say I have the function $f$, in polar coordinates, $$(r,\theta) \rightarrow (r^2,\theta)$$ and I want to find its total derivative at some point $(r,\theta)$ (i.e. its best linear approximation)....
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2answers
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1answer
16 views

Particle traveling on a helix

In this problem, we consider a particle moving on a helix: a) Suppose the position vector of a particle as a function of time $t$ is given by $r(t) = ( \cos t, \sin t, 3t )$. Find the speed of the ...
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1answer
28 views

Does this definition of directional derivative depend on the magnitude?

Here is the definition: My calculus book defines directional derivatives for unit tangent vectors. According to Wikipedia, there is a convention that uses both the direction and magnitude. However, ...
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1answer
20 views

meaning of $\phi$ in Spivak's proof of inverse function theorem

in Spivak's proof of the inverse function theorem, what is the definition of $\phi$? in theorem 2.2: $\phi$ is defined as $\phi(x)=f(x)-f(a)-\lambda (x-a)$. Is the definition of $\phi$ in the ...
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1answer
15 views

Flux across cone

Find the flux of the vector field $F$ across $\sigma$ by expressing $\sigma$ parametrically. $\mathbf{F}(x,y,z)=\mathbf{i+j+k};$ the surface $\sigma$ is the portion of the cone $z=\sqrt{x^2 +y^2}$...
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12 views

Do specific values of partial derivatives allow us to use the chain rule for functions of 2+ variables?

Given a function $f(x, y)$ and knowing that both x and y are functions of $s$ and $t$, can we use the chain rule to find, for example, the partial derivative of f with respect to s ($\frac{\partial f}{...
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1answer
24 views

How to find the function $Y(K,L)$?

$\alpha$ is just a constant and it's given that $$\frac{\partial Y(K,L)}{\partial K}=\alpha\frac{Y}{K}$$ $$\frac{\partial Y(K,L)}{\partial L}=(1-\alpha)\frac{Y}{L}$$ Doing some integration $$\ln(Y)=...
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Integrating a two Dimension function using one dimension delta

I know that given a function $\phi(x)$ and connected open interval $\Omega\in\mathbb{R}$ then \begin{align} \int_{\Omega}\phi(x)\delta(x-b)\;\mathrm{dx}=\phi(b) \end{align} I would like to know if $\...
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1answer
23 views

Changing the order of the integration $\int_{-1}^1\int_0^\sqrt{1-x^2}\int_{x^2+y^2}^1fdzdydx$

I would like to express the integral $$\int_{-1}^1\int_0^\sqrt{1-x^2}\int_{x^2+y^2}^1fdzdydx$$ in the order $dxdydz$. I find $$\int_0^1\int_\sqrt{z}^1\int_{-\sqrt{1-y^2}}^\sqrt{1-y^2}fdxdydz.$$ Could ...
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1answer
23 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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21 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
31 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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0answers
17 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
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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
41 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
177 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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0answers
18 views

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
37 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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2answers
43 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
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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
14 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
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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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3answers
80 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
27 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 ...
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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
25 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
10 views

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
63 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
45 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
32 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
19 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
30 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 ...