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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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10 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
21 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
30 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
15 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
12 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
27 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
39 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
23 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
38 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
20 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
23 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
13 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
60 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
43 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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0answers
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
30 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
13 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
26 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
20 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
22 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 ...
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2answers
154 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) = (\...
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0answers
41 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} \...
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1answer
25 views

Question concerning implicit function theorem

I am studying the implicit function theorem. I have a question about the condition $F_y(x_0,y_0)\neq 0.$ More precisely, let $w=F(x,y)$ be a $C^1$ function on an open rectangle $R=(a,b) \times (c,d)$...
2
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1answer
40 views

Find the volume of the solid bounced by the planes $z=0$,$z=y$ and $x^2+y^2=1$

So I do the following: $$\int_{-1}^{1}\int_0^{\sqrt{1-x^2}} \int_0^{y} \,dzdydx$$, but the answer gives me $\frac{2}{3}$, as it graphs a cylinder it should be the half of the half of a cylinder of ...
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1answer
56 views

Proving that $\lim_{(x,y)\to(0,0)} (x^2+y^2)^{x^2y^2}=1$ using polar coordinates

Am I doing this right? I rewrite the function as follows: $$(r^2\cos^2\theta+r^2\sin^2\theta)^{r^4\cos^2\theta\sin^2\theta} \stackrel{\text{various trig identities}}{=} r^{\frac{1}{4}r^4\sin^2 2\...
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0answers
14 views

How to minimize piecewise binary function with absolute value threshold

Let $f: \Bbb R^n \rightarrow \Bbb R$ and $\mathbf{g}:\Bbb R^n \rightarrow \Bbb R^n$. Let $\mathbf{a,b}$ be random variables with values in $\Bbb R^n$. $$f\mathbf{(a,g(b))} = \begin{cases} ...
4
votes
1answer
38 views

Exponential decay of the gradient if the function itself and the Laplacian have exponential decay

Let $f: \mathbb{R}^n \rightarrow \mathbb{R}$ be twice continously differentiable. For $n=1$ one can prove using Taylors formula that we have $$ \sup_{\vert x \vert \leq R} \vert f'(x) \vert \leq 2\...
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0answers
51 views

How to evaluate the line integral (checking Stokes Theorem)

Consider the vector field: $$\vec F = ye^x \hat i + (x^2 + e^x) \hat j + z^2e^z \hat k$$ A closed curve $C$ lies in the plane $x + y + z = 3$, oriented counterclockwise. The parametric ...
1
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1answer
22 views

Why curl of a vector field that is proportional to $1/r^2$ equal to $0$?

The curl of the vector field ${\bf F} = (-y {\bf i} + x {\bf j})/(x^2 + y^2)$ is $0$. I have an intuitive understanding of why the divergence of a radial field that is proportional to 1/r^2 is equal ...
1
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1answer
15 views

Evaluate the surface integrals using divergence theorem ${\oint \oint}_S (xy\bar{i} + z^2 \bar{k}) \bar{n} dS$. Help finding domain

Evaluate the surface integrals using divergence theorem $$ {\oint \oint}_S (xy\bar{i} + z^2 \bar{k}) \bar{n} dS $$ where S is the surface enclosing the volume in the first octant bounded by the ...
1
vote
1answer
41 views

Use the Stokes's theorem to evaluate the line integral $\oint_c y dx + x dy + (x^2 + y^2 + z^2) dz$

I am using the Stokes's theorem to evaluate the following line integral, $$ \oint_c y dx + x dy + (x^2 + y^2 + z^2) dz $$ where $C$ is the curve $x^2 + y^2 = 1, z = xy$ directed clockwise as viewed ...
0
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0answers
22 views

Prove that $(\mathbf g\circ\mathbf f)_*=\mathbf g_*\circ\mathbf f_*$ and $(\mathbf g\circ\mathbf f)^*=\mathbf g^*\circ\mathbf f^*$

Let $\mathbf f:\mathbf R^n\rightarrow\mathbf R^m$ and $\mathbf g:\mathbf R^m\rightarrow\mathbf R^k$. I figured out the pull-back part by finding $$ (\mathbf g\circ\mathbf f)^*(du_1)=d(g_1(f_1(x_1,...,...