# 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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### 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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### 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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### 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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### 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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### $f$ 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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### 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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### 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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### 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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### 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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### Second derivative test equal to zero for $f(x,y)$ and it has a critical point $(a,b)$, where $f(a,b)$ is a local maximum [closed]

What some examples of functions like these?
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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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.
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,...
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\...