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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0answers
17 views

How to derivate $\mathcal{F}(t,X)=(F(t,x,\lambda),0)$, $X=(x,\lambda)$

So, I have a function $F:\mathbb{R}\times\mathbb{R}^d\times\mathbb{R}^m\rightarrow \mathbb{R}^d$, such that $\frac{\partial F}{\partial x}(t,x,\lambda)$ and $\frac{\partial F}{\partial\lambda}(t,x,\...
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1answer
13 views

Geometric Interpretation of 3D Minimum Distance Using Lagrange

When we try to maximize/minimize the value of $f(x,y) = 2x + y$ on the unit circle $x^2 + y^2 = 1$, we look at the level curves of $2x + y$ and find the $z$ value at which the line is tangent to the ...
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21 views

limit of three variables

I'll glad for a guidance what is the way to determine the next limit (when (x,y,z) goes to (0,0,0) , if is exists? Thanks.
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1answer
21 views

Vector fields and work/flux - calc 3

Here's the image of the problem, it's on work and flux, I'm stuck because I'm not sure if I just need to find the vector between the two points and then find the dot product of it with $f(x,y)\dots$. ...
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2answers
18 views

Example where the partial derivatives exist in a nbhd of a point, but the function is discontinuous at that point

Let $$f\colon \mathbb{R}^n \to \mathbb{R},$$ and assume the partial derivatives of $f$ exist in a neighborhood of $x_0\in \mathbb{R}^n$. We know that in case the partials are also bounded, then $f$ ...
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0answers
6 views

Riemann invariants of a special hyperbolic system

How can one compute the Riemann invariants of the following one dimensional hyperbolic system? $$\begin{pmatrix} u \\ v \end{pmatrix}_t + \begin{pmatrix} -v & -u \\ |v|-k & \mathrm{sgn}(v)u \...
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1answer
13 views

Using mean value theorem for several variables

I found this excercise for a course of calculus in several variables: Show that exists a number $\theta$, $0<\theta<1$ such that $\frac{2}{\pi}=\cos{\frac{\theta\pi}{2}}+\sin{\frac{\pi}{2}(1-\...
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1answer
45 views

Proving $ \idotsint_{A}e^{t\left ( x_1+\ldots +x_n\right )}\,dx_1 \ldots dx_n=\left ( \frac{e^t-1}{t} \right )^n$

Let $$A=\left \{ \left ( x_1,\ldots ,x_n \right )\in \mathbb{R}^n \,\Bigg|\sum\limits_{1\leq k\leq n}\frac{x_k}{k}\leq 1,x_1,\ldots,x_n\geq 0 \right \}$$ Prove that for any $t\in \mathbb{R}$ the ...
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20 views

the curvature as a function of $t$

let $r,c>$ some real numbers, and $X(t)=\langle r\cos t, r\sin t, ct \rangle$ is a curve. Find the curvature as a function of $t$ I think that the curvature of a curve is $$\left\lVert\frac{dT}{ds}\...
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Compute $B_i:\mathbb R^2 \to \mathbb R$ such that $\nabla B(x,y) \cdot A_i(x,y) = 0$ for given $A_i:\mathbb R^2 \to \mathbb R^2$

Let \begin{aligned} &A_{1}=\left(-\frac{x+y \operatorname{sgn}(x)+\sqrt{x^{2}+4 y-4 y|x|+2 x y \operatorname{sgn}(x)+y^{2} \operatorname{sgn}(x)^{2}}}{2(-1+|x|)}, 1\right)\\ &A_{2}=\left(-\...
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1answer
22 views

Partial derivative of vector valued function

I am self-studying vector calculus and trying to differentiate the following function $f$ $$ f(\textbf{t}) = sin(log(\textbf{t}^T\textbf{t})) $$ with respect to $\textbf{t}$ where $\textbf{t} \in \...
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14 views

How can I get the derivative of double integral Lyapunov function?

How can I get the derivative of the following double integral function. This is a commonly used Lyapunov functional to get a delay dependent solution. $V(t) = \tau \int_{-\tau}^{0} \int_{t+\theta}^{t} ...
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2answers
68 views

Multivariable Calculus and Differentiability

$$f(x,y)=\begin{cases}\dfrac{y^3}{x^2+y^2} &(x,y) \neq \ \mathbb{(0,0)}\\ 0 & (x,y)=(0,0) \\ \end{cases}$$ Evaluate $f_x(0,0)$ and $f_y(0,0)$ and $D_\overrightarrow{u}f(0,0)$ I tried directly ...
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20 views

Line Integral of f along close curve and Green's Theorem

I am trying to solve the following question. Let C be a simple closed curve in the plane with positive orientation bounding domain D in the plane. Which of the following integrals below is equal to ...
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38 views

Multivariable function critical points

$$f(x,y)=x\sin(x)+\cos(x)-y\sin(x)+\frac{y^2}{2}$$ My task is to find the critical points of this multivariable function (Determine the set of critical points of the function). Now I found the partial ...
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1answer
25 views

Determining whether $ x^2 + y^2 + 4xy $ can take on negative values

I'm trying to determine whether it's possible for the function $ f(x,y) = x^2 + y^2 + 4xy $ to take on negative values. I know, from looking it up that it can, but how can i show/prove it?
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1answer
44 views

Differentiability of $f : \mathbb{R}^n \rightarrow \mathbb{R}^n$, where $|x-y - (f(x) -f(y))| \leq \frac{1}{2}|x-y|$

Question: If $f : \mathbb{R}^n \rightarrow \mathbb{R}^n$ is a continuous function that satifies $|x-y - (f(x) -f(y))| \leq \frac{1}{2}|x-y|$ for all $x,y \in \mathbb{R}^n$, prove that $f$ is ...
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3answers
76 views

Generalizing dot product to arbitrary order

For a vector $\mathbf{r}$, if we define $\mathbf{r}^1 \equiv \mathbf{r}$ and $\mathbf{r}^2 \equiv \mathbf{r} \cdot \mathbf{r}$, we find that $\mathbf{r}^1$ is a vector and $\mathbf{r}^2$ is a ...
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1answer
66 views

Showing $\lim_{p\to\infty}\left(\int_I|f|^p\right)^{1/p} = \max|f|$, where $I$ is a generalized rectangle

Let $I$ be a generalized rectangle and let $f: I \to \mathbb{R}$ be continuous. Show that $$\lim_{p\to\infty}\left(\int_I|f|^p\right)^{1/p} = \max|f|$$ I found it straightforward to show that these ...
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1answer
40 views

Why differentials can cancel in some versions of chain rule but not others?

This makes perfect sense to me: $$\frac{dy}{dt}=\frac{dy}{dx}\cdot\frac{dx}{dt}$$ This does not: $$\frac{\partial z}{\partial u}=\frac{\partial z}{\partial x}\cdot\frac{\partial x}{\partial u}+\frac{\...
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1answer
14 views

$L^p$ norm of gradient and modulua of the gradient of a real valued function

Let $\Omega$ be a bounded domain over $\mathbb{R}^N$ and $u:\Omega\to\mathbb{R}$ is a function. Then for any $1<p<\infty$, we have $$ \int_{\Omega}|\nabla|u||^p\,dx\leq\int_{\Omega}|\nabla u|^p\,...
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14 views

Examples of strict inclusion of function classes

Consider the following three classes of Lipschitz functions $f:\mathbb{R}^n\to\mathbb{R}^n$: $A=\{\exists\beta,R>0:\,\langle f(x)-f(y),x-y\rangle\leq-\beta|x-y|^2\,\forall|x|,|y|>R\}$ $B=\{\...
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0answers
24 views

$\int_{\partial B_1(0)} \omega f(\omega r) d\omega = 0$ for every $r \ge 0$ implies $\nabla f(0) = 0$?

Let $f:\mathbb R^N \to \mathbb R$ be a smooth function and assume that $$\int_{\partial B_1(0)} \omega f(\omega r) d\omega = 0$$ for every $r \ge 0$, where $\partial B_1(0)$ denotes the sphere of unit ...
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1answer
24 views

Can anyone explain to me this theorem? (Related to the chain rule for vector functions)

Theorem: let $X:I\to \mathbb R^n$ be a curve whose speed $v(t)$ is $>0$ for all $t$ in the interval of definition, Let $\text{ (1)}$ $$s(t)=\int_a^tv(u)\,du \text{ (2)}$$ and $t=f(s)$ be the ...
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1answer
38 views

Non-constant function such that $\int_{\partial B_1(0)} \omega f(\omega r) d\omega = 0$ for every $r \ge 0$?

What are examples of a function $f:\mathbb R^N \to \mathbb R$, other than constant functions, such that $$\int_{\partial B_1(0)} \omega f(\omega r) d\omega = 0$$ for every $r \ge 0$, where $\partial ...
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0answers
11 views

Double Integral Point of Intersection

Should the point of intersection be a solution to all given surface and boundary curves, or can it be an intersection of the two boundary curves only excluding the surface?
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0answers
22 views

Sketch and express a double integral of an arbitrary function

If $f$ is an arbitrary and continuous function in $R$, sketch $R$ and express the following double integral: $$\iint_Rf(x,y) \, dA$$ And $R$ being the region limited by the graph of the following ...
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1answer
40 views

How to choose area of integration for Stokes' theorem on a submanifold of a torus?

This question is coming from physics, so I hope you will be patient as you read the following. Setup: Assume all criteria for applying Stokes' theorem are met. Consider a 2D plane representation of a ...
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1answer
16 views

Problem finding the potential function of a conservative field

I'm trying to find the potential function of this coservative vector field $$F\left(x,y,z\right)=(2xyz^2\ ,x^2z^2+z\cos yz, 2x^2yz+y\cos yz\ ) $$ so I can use the fundamental theorem of calculus to ...
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1answer
23 views

Determine the equation of a a plane tangent at a hyperboloid of one sheet in a point M. Prove that this tangent plane cuts the surface after two lines

Determine the equation of a plane tangent at a hyperboloid of one sheet $\frac{x^2}{4}+\frac{y^2}{9}-\frac{z^2}{1}=1$ in a point M (2,3,1) . Prove that this tangent plane cuts the surface after two ...
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1answer
17 views

How does one find the moment of inertia about a line passing through the centroid of a cone?

Image 1 detailing the question Image 2 showing the part of the working I do not understand In image 2 part d, I'm unsure why the moment of inertia with respect to the y axis, Iy, is equal to Ic plus ...
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0answers
35 views

Find the least positive $x$ such that the expression $a + bx$ is divisible by $n$, where $a$, $b$, $n$ are assigned [duplicate]

I'm trying to solve this general problem(to solve the programming one). We have assigned variables $a$, $b$ and $n$. What is the least positive $x$, so that $a + bx$ is divisible by $n$? It is ...
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0answers
18 views

Volume of a Hyper-sphere in n dimension

Use an n-tuple integral to find the volume enclosed by a hypersphere of radius r in n-dimensional space. [Hint: The formulas are different for n even and n odd.]
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52 views

Proving that inverse of a smooth function is smooth

Suppose I have a smooth function $g: \mathbb{R}^n \to \mathbb{R}^t$ and write the variables as $(x,y)$ where $x \in \mathbb{R}^t$. Suppose the Jacobian matrix of $g(\cdot, y)$ is invertible at $y = 0$ ...
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0answers
19 views

Computing double integral of a 2D vector field

We've received a problem in my mechanics class, out of Taylor, to calculate the center-of-mass of a uniform, thin semicircle of metal with radius $R$ and diameter along the $x$-axis. Here's my ...
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1answer
24 views

Derivative of Unit Speed Parameterization of a Curve

I've been working on this question for about 3 hours now. Part (a) asks to show that the derivative of the unit speed parameterization function is perpendicular to its second derivative. As far as I ...
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2answers
35 views

Prove that $\forall x\in E, \exists z\in F\subset E,\; \inf_{y\in F}\|x-y\|=\|x-z\|$, where $F$ is compact and $E$ is a normed vector space.

Let $(E,\|\cdot\|)$ be a normed vector space and let $F\subseteq E$ be a non-empty compact subset. We define the distance from $x$ to $F$ as $$\begin{align} d\colon &E\times\mathcal{P}(E)\setminus ...
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1answer
32 views

Change to polar coordinates and integrate.

Find the integral $$\int_{-\sqrt{3}}^{\sqrt{3}} ~dx \int_3^{2 + \sqrt{4-x^2}} \frac{~dy}{(x^2 + y^2)^\frac{3}{2}}$$ by changing to polar coordinates. I've managed to find the polar coordinates. The ...
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0answers
16 views

Line integral of a vector field over a intersection of a cylinder and a plane

I need help calculating the line integral of this vector conservative field $F(x,y,z)=(2xyz^2, x^2z^2+zcos(yz), 2x^2yz+ycos(yz))$ Compute $\oint_{c}^{} Fds$ Where C is the intersection of the cylinder ...
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0answers
22 views

Stokes' theorem problem, how to find sigma

Problem: If 𝐹⃗(𝑥, 𝑦, 𝑧) = (3𝑧 − sin𝑥)𝑖̂+ (𝑥^2 + 𝑒^𝑦)𝑗̂+ (𝑦^3 − cos𝑧)𝑘̂, use Stokes’ theorem to find the counterclockwise circulation where C is the curve 𝑥 = cos 𝑡, 𝑦 = sin 𝑡, 𝑧 = 1;...
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20 views

Divergence functions [closed]

Please help me in solving it and guide me i totally have no idea how to solve it.
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0answers
19 views

Divergence and curl integral surfaces

Show that if $f=0$ on the unit sphere $r=1$ then $$f(0,0,0) = \lim_{\epsilon \to 0}\frac{-1}{4\pi}\iiint_{D_\epsilon}\frac{(x,y,z)\cdot\nabla f}{r^3}\,dV,$$ where $D_\epsilon = \{\epsilon \leq r \leq ...
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1answer
16 views

Volume bounded by the paraboloid and plane

Determine the volume of the solid $D$ that is bounded by the paraboloid $z = a(x^2 + y^2)$ and the plane $z = b$ in terms of $a$ and $b,$ where $b > 0, a > 0.$
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0answers
23 views

Integration by parts on the boundary of a Lipschitz domain in $\mathbb{R}^2$

Let $\Omega\subset\mathbb{R}^2$ be a bounded and connected Lipschitz domain. What are the least restrictive conditions that real-valued functions $f$ and $g$ must satisfy in order for the condition $$\...
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1answer
26 views

Proof of directional derivatives must vanish at bistable points.

I am self learning optimization from the book "Non-convex Optimization for Machine Learning". I stumbled upon the below statement, "Directional derivatives must vanish at bistable ...
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0answers
20 views

Kindly see the question in the picture. [closed]

enter image description here kindly open the link for the question and please provide a solution.
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1answer
53 views

$\lim_{(x,y) \to (0,0)} (x^2+y^2)\tan\left(\frac{\pi}{2+x^2+y}\right)$ does not exist.

I need to prove this limit does not exist. I already tried to change to polar coordinates and got $\lim_{r \to 0} r^2\tan\left( \dfrac{\pi}{2+r^2\cos^2\theta+r\sin\theta}\right)$, which I tried to ...
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0answers
26 views

It there other method to show the function $t\mapsto \int_0^{\infty}\frac{\cos x}{1+(x+t)^2} dx$ is differentiable for all $t\in\mathbb{R}$

In order to show that the function $F\colon t\to \int_0^{\infty}\frac{\cos x}{1+(x+t)^2} dx$ is differentiable for all $t\in\mathbb{R},$ I make a change of variable of $T\colon ~y=x+t,$ so that \begin{...
2
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1answer
98 views

Integrating in Polar Coordinates

Calculate the following integral in the following region: $$Q_r=\lbrace (x,y) \in \mathbb{R}^2: -r \le x,y\leq r\rbrace$$ $$\iint_{Q_r} e^{-x^2-y^2} \mathrm{d}x \mathrm{d}y$$ Since the region where ...
4
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3answers
96 views

Double Integral $\iint e^\frac{x+y}{x-y} \,dx \,dy$ solution

In the Multivariable Calculus Class I'm taking we were tasked to solve $$\iint_{R} e^\frac{x+y}{x-y} \,dx \,dy\,,$$ where $R$ is the trapezoidal region with vertices $(1,0)$, $(2,0)$, $(0,-1)$ and $(0,...

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