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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1answer
26 views

How to find a useful variable change for this integral

I would like to find the area of the following region $$ D=\left \{(x,y): -\sqrt{1+y^2}\leq x\leq \sqrt{1+y^2}; -1\leq y\leq (x+1)/2\right \}. $$ I try to calculate the double integral brute force, ...
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17 views

Notation for directional derivative

I know if $\vec u$ is a unit vector, then $\triangledown(x,y) \bullet \vec u$ is the directional derivative, or $D_{u}(a,b)$ However, if $u$ is a vector, when $f_{\vec u}(a,b)$ is a real, what does it ...
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Show that $\frac{\delta L}{\delta v^2}$ be constant?

Given $$2 \vec{\varepsilon }\cdot\frac{\delta L}{\delta v^2}\frac{d \vec{r}}{dt} = \frac{dg(\vec{r},t)}{dt}$$ with $\vec{\varepsilon }\in \mathbb{R}^3$ is infinitesimal small vector, $\vec{r}=\vec{r}(...
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2answers
29 views

Continuity problem $f(x,y)=\frac{\ln(1+x^3+y^3)}{|x|+|y|}$

The problem is to check if function: $f(x,y)=\frac{\ln(1+x^3+y^3)}{|x|+|y|}$, when $(x,y)\neq (0,0)$; $f(x,y)=0$ when, $(x,y)= (0,0)$ is continuable or not continuable in point $(0,0)$. I thought of ...
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1answer
17 views

Work done by the force $F(x,y,z)=(y-z)i+(z-x)j+(x-y)k$

I need to compute the work done by the force $F(x,y,z)=(y-z)i+(z-x)j+(x-y)k$ on C such that C is the intersection between $x^2+y^2=4,~x+2z=2$. $rot(F)=(-2,-2,-2)$, then I applied Green's Theorem and ...
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16 views

Directional derivative: is it correct?

Given the following function $$f(x,y)=2x^3y^4$$ I gotta evaluate $D_{\vec{v}}f(1,-1)$, where $\vec{v}=(-3,4)$. I know $$\nabla f(x,y)=(6x^2y^4,8x^3y^3)$$ and so $$\nabla f(1,-1)=(6,-8)$$ Also, the ...
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1answer
21 views

Basics on Vector Calculus

The vorticity equation of fluid dynamics is written as \begin{align} \dfrac{\partial\boldsymbol{\omega}}{\partial t}+(\mathbf{u}\cdot\nabla)\boldsymbol{\omega}=(\boldsymbol{\omega}\cdot\nabla)\mathbf{...
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1answer
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Triple Integral / Volume Evaluation

Question: Find the volume of the solid that lies within both the cylinder $x^2+y^2=1$ and the sphere $x^2+y^2+z^2=4$. The solution given to us by our instructor: $$\int_{0} ^{2\pi}\int _{0} ^{1} \int ...
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Can we differentiate L2-norm? [closed]

For solving an optimization problem, I often see an objective function including$-$ $$||\mathbb{a}||_2=\sqrt{a_1^2+...+a_n^2}$$ I wonder how to differentiate it. That is, my question is: Can you ...
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1answer
47 views

Integral of sin(|x-y|)

I do not know how should I proceed to calculate the definite integral: $$\int_{\frac{-\pi}{2}}^{\frac{\pi}{2}}\int_{\frac{-\pi}{2}}^{\frac{\pi}{2}} \sin(\mid{x-y}\mid)dxdy$$
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Expressing Lipschitz function similar to the mean value theorem.

Let $f(x):\mathbb{R}^n\rightarrow\mathbb{R}^n$ be a globally Lipschitz function. Does this mean that there exists a matrix $M(x,y)$ such that: $$f(x)-f(y)=M(x,y)[x-y]$$ I do not want to use the ...
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How to use the Lagrange method to find the conditional local extrema for the function $f(x,y,z)=x+y+z$?

The conditions are: $x^{2}+y^{2}+z^{2}=5$ and that $x+2y=0$. On my own I've figured out that the first condition is comparable to the distance-from-point formula for multi-dimensional space (it's that ...
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29 views

Trivariate entropy

Entropy is usually defined for the univariate case only, $$H(X) = -\int_{-\infty}^{\infty} f(x) \ln f(x) \hspace{1mm} dx$$ For bivariate copula, bivariate entropy is commonly applied. $$H(X,Y) = -\...
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Describing $\frac{\partial}{\partial x} \oint_{\partial \Omega(x)} f(x, n) \; \mathrm{d}n$ as a contour integral.

My question essentially has to do with the derivative of a Contour Integral's parameterized curve. $$\frac{\partial}{\partial x} \oint_{\partial \Omega(x)} f(n, x) \; \mathrm{d}n$$ to be exact. Where $...
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Implicit function graphing

I came across the function, $$z=\frac{4xy(x^2-y^2)}{x^2+y^2}$$ and I had to draw the level curve for this equation. Clearly the equation is already explicitly expressed in the form $z=f(x,y)$ and ...
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Can someone help in solving these 3 math questions in the picture?

These are three math questions related to derivatives inside this picture I tried to solve the first two questions and I need to know If my answer is right or not and I couldn't solve the last one. ...
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Proving the non existence of a multivariable limit.

For the problem : A part of the solution was given as : We now claim that $\lim\limits_{(x, y) \to(0,0)} f(x, y)$ does not exist. Assume to the contrary that $\lim\limits_{(x, y)\to(0,0)} f(x, y)$ ...
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Probability Distribution constant for equation derivation

I need to develop a formula , may be with some constant derived, based on output i get with three independent variables (Number of patients (N), bacteria generation rate per second (p) ,interval I (...
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15 views

Best approximation of the Mahalanobis distance by standardized Euclidean distance

I am looking for the best way to approximate the Mahalanobis distance by the standardized Euclidean distance, which would reduce the number of the required multiplications. The easiest way is the ...
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1answer
40 views

$f$ have finitely many critical points in $\Omega$

Assume that $\Omega$ is an bounded open set in $R^m, f\in C^2(\overline{\Omega},R^m)$. If $f$ does not have any critical point in $\partial \Omega$, and all the critical points of $f$ in $\Omega$ are ...
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1answer
39 views

Evaluating $\int_{-4} ^4\int _0 ^{\sqrt{16-x^2}} \int _0 ^{16-x^2-y^2} \sqrt{x^2 + y^2}\,dz\,dy\,dx$

Question: Evaluate the given triple integral with cylindrical coordinates: $$\int_{-4} ^4\int _0 ^{\sqrt{16-x^2}} \int _0 ^{16-x^2-y^2} \sqrt{x^2 + y^2}\,dz\,dy\,dx$$ My solution (attempt): Upon ...
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26 views

When to keep a variable constant

I have been wondering when exactly I am allowed to hold other variables constant and when I can’t for integration like $F(x, y)$ with respect to $x$. If the answer is whenever $x$ and $y$ are NOT ...
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32 views

The proof of the product of two continuous functions is also a continuous function in multivariable case.

I want to show that the the product rule of continuous function can be extend to the multivariable case. So, by my setting, given two functions $f(x)$ and $g(y)$ are both continuous function defined ...
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2answers
18 views

In what situation a partial derivative doesn't exist

I am very aware that a partial derivative could be not continuous at a certain point. However, I don't understand when can a partial derivative doesn't exist. Because on the book, they always say &...
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2answers
30 views

subjects to 4 constraints problem on Karushn-Kuhn-Tucker

I will solve the maximization problem, but my constraints are $y \leq 0.5x^2$, $x \geq 0$, $y \geq 0$, and $x \leq 1$. I am wondering that how can I solve the problem because I am wondering that ...
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1answer
59 views

Prove that $|f(x)-f(y)| \geq k|x-y|$

Let $f:R^m \rightarrow R^m$ be $C^1$. And there exists $k>0$ s.t. $\forall x,h \in R^m$, $h^{T}f'(x)h \geq k|h|^2$. Prove that $\forall x,y\in R^m$, $|f(x)-f(y)| \geq k|x-y|$. Well, if $m=1$ then $...
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30 views

Can we solve this integral in terms of whatever form of functions.

\begin{equation} \label{1} \begin{aligned} I_{N} &= \int_{X_{ \ell}} \prod_{i=1}^{N}\left[ \left(\sum_{\ell=1}^{2m_{i}}x_{\ell}^{2}\right)^{k_{i}} \exp{\left(-h_{i}\sum_{\ell=1}^{2m_{i}} x_{ \...
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1answer
11 views

Partial derivatives and the chain rule in $\frac{\partial^2 F}{\partial s^2}$ with $F(s,t)=f(s^2-t^2,ast)$

I'm trying to calculate $\frac{\partial^2 F}{\partial s^2}$ for an exercise and I'm a bit confused. The function given is $F(s,t)=f(s^2-t^2,ast)$. This is the correction: $\frac{\partial^2 F}{\partial ...
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2answers
22 views

Partial Differential Calcuration

$f:R^2 \rightarrow R,\\f(x,y)=\exp(x^2+xy+y^2)$ $D_1:=\dfrac{\partial}{\partial x}$, $D_2:=\dfrac{\partial}{\partial y}$ Can I calcurate $D_1D_2f(0,0):= \dfrac{\partial ^2 f}{\partial x \partial y}(0,...
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0answers
17 views

Setting up an irated integral?

Using whatever method you like, evaluate the integral $I =\int \int \int_ExdV$ where $E$ is the region between the cylinders $x^2+y^2=1$ and $x^2+y^2=4$, below the plane $z=y+2$ and above the $xy$-...
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1answer
25 views

How would I find the minimum of the sum of $x, y, z$, given the product that $xyz=27$ using Lagrange Multiplier?

Find three positive integers x, y, z that satisfy the given conditions. The product is 27, and the sum is a minimum. I'm lost on how I would solve my system of equations that I have set up. $1=\lambda ...
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0answers
57 views

Proving $\int \int _Rf\left(\sqrt{\ \frac{x^2}{3}+\frac{y^2}{5}}\right)dA=2\pi \sqrt{\ 15}\int _0^1\ f\left(p\right)pdp$

Where where f is a continous function on $[0,1]$ and The region R is bounded by the ellipse $5x^2 + 3y^2 = 15$ I'm confused how to approach this, I just recently started with double integration and i ...
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0answers
13 views

Is this function homothetic?

I need to check whether the following function is homothetic or not: f(x,y)=x3y6+3x2y4+6xy2+9 for x,y ∈ R+ As it can be clearly expressed as a positive monotonic transformation of the homogeneous ...
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1answer
31 views

Problem proving a multivariable limit using sandwich theorem

I am asked to prove that the following limit exists: $$\displaystyle \lim_{(x,y) \to (0,0)} \frac{x^2+y^2-x^2y^2}{x^2+y^2} =1 $$ I am able to deduce that the limit exists and equals 1 by trying a few ...
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1answer
29 views

Can I use $y=r\cos(\theta)$, $z=r\sin(\theta)$, and $x=x$ as cylindrical coordinates?

I need to compute $\displaystyle\int_{-\sqrt{2}}^{\sqrt{2}}\displaystyle\int_{-\sqrt{2-z^2}}^{\sqrt{2-z^2}}\displaystyle\int_{0}^{1+y^2+z^2}z^3dxdydz~$ using cylindrical or spherical coordinates. ...
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1answer
37 views

Prove $\int_{|k| < 1} \frac{dk}{|k|^2} \lt \infty$ only when the dimension $d$ of $k$ is $ d \ge 3$

Let $k$ be a vector of dimension $d$. Let $|k|^2 = k_1^2 + k_2^2 + ... + k_d^2$. How can I prove that $\int_{|k| < 1} \frac{dk}{|k|^2}$ is $ \lt \infty $ only when $d \ge 3$?
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1answer
46 views

How to find the linearization of this function($ x^4+y^3-x^3-y^2+x^2+y-2=0$)? [closed]

This past paper question about linearization for my calculus 1 course has been bothering me. I spent 5 days trying to solve it. Considering the equation: $ x^4+y^3-x^3-y^2+x^2+y-2=0$ We accept and can ...
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0answers
15 views

What's the general procedure to finding these directional derivatives for a piecewise function?

What exactly does $D_1$ and $D_2$ denote here? I'm somewhat confused about the definition of these directional derivatives. Would $1$ signify the vector $(1,0)$ and $2$ the vector $(0,1)$?
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1answer
27 views

Find a function $f$ such that $f$ is discontinuous but $W(f,x)$ is a continuous function

I need some help Find a function $f$ such that $f$ is discontinuous but $W(f,x)$ is a continuous function (where $W(f,x)$ is the oscillation of f in a point, this means that $W(f,x)=inf[M_\delta(f)-m_\...
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1answer
46 views

How do I find the bounds of integration for a triple integral?

How do I find the bounds of integration for this? I assume I need to find the standard equation for this so I put it like so: $$\frac{x}{7}+\frac{y}{9}+\frac{z}{2}=1$$ and so I solve for $z$. Then I ...
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1answer
37 views

Find $\int_0^2 \int_0^{\sqrt{3}x} f(\sqrt{x^2+y^2})dydx$ in polar coordinates.

I need to find: $$\int_0^2 \int_0^{\sqrt{3}x} f(\sqrt{x^2+y^2})dydx$$ in polar coordinates. Since $x=r\cos(\theta)$ and $y=r\sin(\theta)$, I got: $y=\sqrt{3}x \iff r\sin(\theta)=\sqrt{3}r\cos(\theta)\...
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1answer
9 views

Finding the product of the slopes of 2 unknown lines tangent to a circle with known intersection point outside circle

Given, two lines are tangent to the circle, and they intersect at a point $(0,14)$ not on the circle, find the product of the slopes $m_1*m_2$ $$l_{1}: y = m_{1}x + c_{1}$$ $$l_{2}: y = m_{2}x + c_{2}$...
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1answer
30 views

Finding the minimum cost

I am currently trying to formulate a formula for this problem using Multivariable Calculus. There is a job of shipping $V$ cubic feet of old steel shavings across the river by barge. To do this, you'...
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2answers
33 views

Finding the bounds of a triple integral (spherical coordinates)

I'm currently learning how to calculate the volume of a 3D surface expressed in spherical coordinates using triple integrals. There was this exercice (from here) which asked me to find the volume of ...
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0answers
15 views

How can I calculate the sub-gradient of this function?

I am facing difficulty in arriving at the sub-gradient of the following function $f(\theta) = \sum_{k = 1}^M | y_k - \theta^T x_k | + \lambda \sum_{k=1}^M |\theta_k|$ Here $x_k $ is $n\times 1$ ...
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2answers
39 views

Surface integral over a cylinder problem

I have a generic cylinder $x^2+y^2=a^2, 0≤z≤h$ labeled as $G$. It has an outward-pointing unit normal vector $\vec{n}$. As seen in the picture below: Now I am asked to solve this integral on $G$: $$\...
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0answers
13 views

Original function from total differential

Given total differential dg=ydx-xdy, find a function g(x,y) that satisfies the total differential. Any such function is sufficient.
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24 views

Calculate the area between curves on unit sphere

Pardon me if someone else have asked this. If we have 2 curves $r_1(t), r_2(t)$$: [0, 1] \to S^2$, (either we know the exact equations or not) on the unit sphere such that their endpoints are the same,...
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1answer
26 views

Definition of limits at infinity in higher dimensions using sequences

Im studying multivariable calculus and stumbled up an alternate way to talk about limits at infinity since one dimensional definition is of no use. So for $\mathbb{R^n}$ with $n \geq 2$, we let $\{a_n\...

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