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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0
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1answer
30 views

Find the area bounded by a curve by changing variables

Calculate the area bounded by the following formula: $$\left(\frac{x^2}{a^2}+\frac{y^2}{b^2} \right)^2 = \frac{xy}{c^2}$$ where $a,b,c>0.$ I have used changing variable of $x=au$ and $y=vb$ to ...
0
votes
4answers
73 views

Where is $|xy|$ function differentiable

I'm trying to solve this problem: Let $f(x,y) = |xy|$. Find the sets of all points $(x,y) \in \Bbb R$ where $f$ is differentiable and compute the differential in those points. Can someone explain ...
2
votes
4answers
105 views

How does one parameterize $x^2 + xy + y^2 = \frac{1}{2}$?

Parameterize the curve $C$ that intersects the surface $x^2+y^2+z^2=1$ and the plane $x+y+z=0$. I have this replacing equations: $$ x^2+y^2+(-x-y)^2=1$$ and clearing have the following: $$ ...
5
votes
3answers
75 views

Can the directional derivative fail to be linear?

Is it possible for the directional derivative for a function $f$ in the direction of a vector $v$, $D_vf(x) = \lim_{h \to 0} \frac{f(x + hv) - f(x)}h$ to exist for every vector $v$, and yet $v \mapsto ...
0
votes
1answer
36 views

How to use Implicit Function Theorem for this function?

Let $x = (x_{1}, x_{2}, x_{3})^{T} \in \mathbb{R}^{3}$, and take a sufficiently differentiable function $f: \mathbb{R}^{3} \mapsto \mathbb{R}$. Suppose that I am searching for zeroes of $f$ such ...
2
votes
1answer
40 views

Can we find a regular ($C^k$) parametrization for this surface?

I have here a surface whose curvature properties I want to study, represented in cylindrical coordinates: $$f(r,\theta) = r^2\cos4\theta$$ The problem, however, is that the parametrization is not ...
1
vote
3answers
28 views

Show that $1/|x|$ is not Lipschitz continuous on $|x|<1$.

$x$ is a $3$d vector. This is what I have so far, don't know if it is enough to prove , that $f(x) = 1/|x|$ is not Lipschitz-continuous on $|x|<1$: First we have to show, that for all $L>0$ ...
1
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1answer
38 views

Triple integral vs double integral to find volume of an object

Is it possible to find the volume of an object bounded by two surfaces in both of these two ways?: -a triple integral of 1 dV (I know this works) -a double integral of the top surface - bottom ...
0
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0answers
15 views

Gaussian region in $\mathbb R^n$

My question pertains to the paper "A Simplified Proof of the Divergence Theorem" by Djairo Guedes de Figueiredo. The paper says: A Gaussian region is an open connected bounded set $V$ in $\mathbb ...
6
votes
3answers
531 views

Why is this set not a manifold?

Set $M = \{ \, (x, y) : x^2 = y^2 \, \}$. If for every point $(a, c)$ in $M$, there exists a neighborhood $U$ containing $(a, c)$ and function $\phi(x, y)$ such that: $\phi(x, y) = 0$ on $M \cap ...
0
votes
1answer
15 views

Estimating the curvature of a discretized curve in 3d with cubic splines

I have a computer simulation in which I'm modeling a physical curve by discretizing it and updating the locations of these points. I want to find/estimate the location of the maximum curvature of the ...
0
votes
2answers
23 views

Show that $f(x) = x \cdot |x|^2$ with $|x|<1$ is Lipshitz continuous.

I am reading Body & Soul, Part $3$ and got stuck with this exercise: Show that $f(x) = x \lvert x\rvert^2$ with $\lvert x\rvert<1$ is Lipschitz continuous, where $x$ is a $3$d vector. I ...
2
votes
1answer
20 views

Total Derivative at a point

Let $f(x,y)=x^{3}+y^{3}$ . How do I find the total derivative at (0,0) by using the definition? I am confused of what to take as the Error function.
4
votes
1answer
33 views

Integrating both sides of an equation with respect to different variables [duplicate]

So im reading a book called "Ordinary Differential Equations" (Tenenbaum & Pollard) and in the introduction(ish) they are doing an example using a carbon dating problem, represented as: ...
0
votes
0answers
39 views

The most general way to prove differentiability over an interval

Preface: I'm going to try and make this question as general as I possibly can, as there are many different extensions of Calculus (Single-Variable, Multi-Variable, Vector, Tensor etc.), in which ...
0
votes
2answers
40 views

Prove that $\lim \limits_{(x,y)\to (c,0)} \frac{\sin(x^2y)}{x^2-y^2}=0$, $c\ne0$ with (ε, δ)-definition

So the problem is to work with $\left|\frac{\sin(x^2y)}{x^2-y^2}\right|$ and show that that is less than a formula involving δ, let´s call it g(δ), formula which I will later equal to ε in order to ...
1
vote
1answer
27 views

Solving a multivariate polynomial system involving the power sums

I would like to know if there is a way to solve or simplify the system of equations given by: $$ x_1^1+x_2^1+\cdots x_n^1 = c_1\\ x_1^2+x_2^2+\cdots x_n^2 = c_2\\ \vdots\\ x_1^n+x_2^n+\cdots x_n^n = ...
1
vote
1answer
28 views

Can a two-variables function be an odd function in one variable?

Like in single variable, we use $f(-x)=-f(x)$ to show that a function is odd. Similarly, for two variables, we can use $f(-x,-y)=-f(x,y)$. If we have a two variable function like this ...
0
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0answers
28 views

How to calculate partial derivative of many values?

I have a function for example : $$f(x,y) = x^5 + 3xy + \cos(xy)$$ It's easy to calculate the partial derivative of $x$ or $y$. But how to calculate the partial derivative of $x$ AND $y$, $[ f'x,y ]$
1
vote
1answer
33 views

Encountered a problem with a smaller integral within the bigger double integral

The question states to solve the following equation by transforming to polar coordinates.$$\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} e^{-(x^{2}+y^{2})} \ dx \ dy$$ So far I have managed to put ...
0
votes
0answers
18 views

Help with limits of multivariable function?

I need help with understanding part of a proof: I am told that I have $f:\mathbb{R}^{2}\rightarrow \mathbb{R}$ such that: $\lim_{(x,y)\rightarrow (a,b)}f(x,y)=l$ and for all $x$ in $\mathbb{R}$: ...
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votes
2answers
54 views

Without Extreme Value Theorem, how do we find absolute extrema?

I have to find and classify the critical points of the following functions and then state which relative extrema are absolute extrema. $$f(x,y) = x^3 - y^3 - 2xy + 6$$ $$f(x,y) = xy + 2x - ...
1
vote
0answers
53 views

On a function $f: \mathbb R^m \to \mathbb R^n$ , $n>1$ , mapping connected sets to connedted sets and discontinuous at a point

Let $f: \mathbb R^m \to \mathbb R^n$ be a function mapping connected sets to connected sets where $n>1$ ; let $a \in \mathbb R^m $ and $ \epsilon >0$ be such that $f(B_{\delta}(a)) \cap ...
0
votes
1answer
29 views

Where does lambda come from in Lagrange multipliers? Specifically related to find maximum and minimum values on a constraint.

The book states that the $\nabla f(x,y,z) = \lambda\nabla g(x,y,z)$ It talks about the slope of the tangents being parallel, but wouldn't they technically just be same line? It also brings up norms ...
-2
votes
0answers
32 views

Help finding the maximum and minimum values using lagranges method [duplicate]

Where $f(x,y,z) = xyz$ and the constraint is $g(x,y,z) = x^2+2y^2+3z^2 = 6$ after solving Equation 1 $=$ $\nabla$$f_x$ $=$ $\lambda$ $\nabla$$g_x$ $=$ [ $yz$ $=$ $\lambda$2$x$] for x and obtain: $x$ ...
0
votes
1answer
80 views

Find the Lagrange multipliers with one constraint: $f(x,y,z) = xyz$ and $g(x,y,z) = x^2+2y^2+3z^2 = 6$

Where $f(x,y,z) = xyz$ and the constraint is $g(x,y,z) = x^2+2y^2+3z^2 = 6$ I have tried this problem like three or four times and not gotten the solution, I even asked this question once and got the ...
2
votes
1answer
18 views

Derivative of the stress tensor

Let $\partial u_i/\partial x_i=0$ then given that $$\sigma_{ij} = -p\delta_{ij}+\mu\left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}\right)$$ Show that $$\frac{\partial ...
1
vote
2answers
22 views

extremum under constraint question

I was tasked with finding the extremum of $z=xy$ under the constraint $x+y=1$, here is what I did: $$z=xy$$ $$x+y=1$$ from the second line we get $y=x-1$ and we substitute that back in the first ...
3
votes
0answers
29 views

Question about Gauss divergence theorem

The Divergence Theorem says that for $\Omega \subset \mathbb{R}^n$ $$\int_\Omega \nabla \cdot F = \int_{\partial\Omega} F \cdot n $$ where $n$ is the outward normal. Assuming that $F$ is a smooth ...
0
votes
2answers
48 views

How do I prove that compositions of $C^\infty$ functions are $C^\infty$?

Let $G$ be open in $\mathbb{R}^n$. Let $f:G\rightarrow \mathbb{R}$ be a $C^\infty$ function and $g:\mathbb{R}\rightarrow \mathbb{R}$ be a $C^\infty$ function. Then, how do I prove that $g\circ f$ is ...
2
votes
1answer
32 views

Evaluate the volume of the solid defined by $x^2+y^2+z^2 \leq 9$ and $x^2+y^2 \leq 3y$

I am asked to solve the following problem: Evaluate the volume of the solid defined by $x^2+y^2+z^2 \leq 9$ and $x^2+y^2 \leq 3y$. I thought about using spherical coordinates: $$ 0 \leq \rho ...
0
votes
1answer
19 views

Evaluate $f(x,y,z) = z^3$ on the region defined by $z \geq 0 \ x^2+y^2 \leq 1 \ x^2+y^2+z^2 \leq 2$

I am asked to solve the following problem: Changing the variables, evaluate the integral of the function $f(x,y,z) = z^3$ on the region defined by $z \geq 0 \quad x^2+y^2 \leq 1 \quad ...
0
votes
1answer
21 views

Related Quantity Differentials

My book has the following question I have no idea how to solve. It is the only problem I don't understand in the chapter, I want to have an answer before I move on. The question is: For related ...
0
votes
2answers
29 views

Double integral over region $R$ using change of variables

Can someone help step by step on the following? Evaluate $\iint_R x^2y^4\,dxdy$, where $R$ is the region bounded by $xy = 4$, $xy = 10$, $y = x$, and $y = 6x$ by using the change of variables ...
0
votes
3answers
30 views

Continuous for each variables does not implies continuous

Prove or disprove the following statement: Statement. Continuous for each variables, when other variables are fixed, implies continuous? More clearly, prove or disprove the following problem: Let ...
0
votes
1answer
29 views

Triple Integration of gravitational potential

Integrate $\int_{0}^{2\pi } \int_{0}^{a} \int_{0}^{\pi/2} \frac{ G\rho r^2 sin \theta}{ {(r^2-2rt cos \theta + t^2 )}^{\frac{1}{2}}} d \theta dr d \phi$, where $\rho, t $ are constants. Sorry ...
0
votes
3answers
28 views

Showing a function of 2 variables is differentiable over ${R^2}$

I am self studying multivariable calculus and I want to know how to show that $f(x,y) = {x^2} + {y^2}$ is differentiable. At least that is the first example from my book, if you know a more ...
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votes
0answers
16 views

Differentials of related quantities…

I am self studying multivariable calculus and there is a question in a book that I don't understand. It's because I'm not really clear what a differential is versus a derivative. I don't need to be ...
0
votes
2answers
16 views

extremum under constraint

I had to find the extremum of $z=x^2+y^2$ under the constraint of $x+y=3$; I used Lagrange multipliers to reach the conclusion that $(1.5,1.5)$ is an extermum point, but had no way of determining ...
1
vote
2answers
43 views

Exact vs. conservative

I'm having trouble understanding definitions. What's the difference between something being exact and being conservative? I understand both involve proving that a potential function $f$ exists such ...
0
votes
1answer
34 views

multivariable maxima and minima

I try to calculate minma and maxima in $ x^2 + y^2 -12x+16y = z$ for the boundary $x^2+y^2<=25$ however I keep getting only$ (6,-8)$ as a possible point when there are other two answers . how do ...
1
vote
2answers
21 views

Find the extreme values of $f(x,y)=xy$ on $D=\{(x,y)|1 \leq x^2+y^2 \leq 4\}$

This would have to be done using conditional extremes(Lagrange method), and maybe some topological properties.I do not know how to do this, I have only done cases where the $D$ would be defined with ...
0
votes
1answer
28 views

Help showing that u is harmonic. [duplicate]

I know that u is a continuous function where $$u(x_0) $$ is equal to the average over the surface of all balls centered at a point $x_0$. How would I show that this function is harmonic, or ...
2
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2answers
25 views

How to solve this double integral involving trig substitution (using tangent function)?

This is a question I came across and I cannot find the answer. By using a substitution involving the tangent function, show that ...
3
votes
3answers
46 views

How to take the “gradient” of a matrix?

I want to find $(D^2 F)$ where $\vec{F}(\vec{x}) = \frac{\vec{x}}{\|\vec{x}\|}$ ($F$ is a row vector and $Du$ is a "column vector", where $u\in \mathbb{R}$ ). I know that $$(DF)_{ij} = ...
1
vote
1answer
71 views

Method to find the extremal values of $xyz$ subject to $x^2+2y^2+3z^2=a$

This question has been asked before but I want to lay out my method and get feedback on reasoning and process this took me a long to put together as I am new to the formatting: Let the function $f$ ...
0
votes
0answers
36 views

$\beta(s) = \int_0^s B(u)\,du$ is a unit speed curve

I'm teaching myself in this differential geometry book, so please have some patience with me if I ask something obvious. Let $\alpha(s)$ be a unit speed curve with domain $(-\epsilon,\epsilon)$ and ...
2
votes
1answer
21 views

Difficulties on proving the continuity part of a homeomorphism

I am trying to prove that the open unitary disk $\mathbb{D}^n$ is homeomorphic to $\mathbb{R}^n$, so the way i am doing it is by showing that the function $$f(x)=\frac{1}{1-|x|}x$$ where $x$ is a ...
1
vote
2answers
17 views

Real-valued function defined by the Cartesian basis vectors…

If ${e_x}$ and ${e_y}$ denote the unit Cartesian basis vectors and for a real-valued function $f(x,y)$, $$\nabla f = ({{\delta f} \over {\delta x}},{{\delta f} \over {\delta y}}) \equiv {{\delta f} ...
0
votes
1answer
23 views

Calculating Flux of F across G with paraboloid and plane

Calculate the flux of F across G where $\mathbf F(x, y, z) = 6x\mathbf i + 6y\mathbf j + 2\mathbf k$; G is the surface cut from the bottom of the paraboloid $z = x^2 + y^2$ by the plane z = 3 I found ...