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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2
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2answers
39 views

Conditional Extremes/Lagrange multipliers: proving: $\frac{1}{x_1}+…+\frac{1}{x_n} \geq \frac{n^2}{x_1+…+x_n}$

$$\frac{1}{x_1}+...+\frac{1}{x_n} \geq \frac{n^2}{x_1+...+x_n};x_i>0.$$ This is supposed to be proven using conditional extremes. I tried the main function being ...
0
votes
0answers
19 views

Does the exchange of differentiation and integration hold?

Let $d\geq 2$, and $\alpha\in(0,2)$. Does the following equation hold? $$\frac{d}{dt}\int_{\mathbb R^d}e^{-it^{-1/\alpha}(x,z)-|z|^{\alpha}}dz=\int_{\mathbb ...
0
votes
0answers
21 views

Finding local extrema of implicit function

Is finding local extrema values of function the same as finding local extrema of implicit function $y(x)$? For example is the method for finding this: $f(x,y)=3x^2y-x^3-y^4$ Same as $x^2 y^2 -x^4 + ...
1
vote
1answer
23 views

A Question on First Order Exact Differential Equations.

As I learned , an exact differential equation is given by the following form $A(x,y)+B(x,y)\frac{dy}{dx}=0$ Following an algebric manipulation it is possible to reach the following : ...
0
votes
0answers
12 views

Infinite roots for $f:R^2 \to R$ continuous on a Ringed Domain.

I need help formalising my idea for this problem: Given the domain $ D = \{(x,y) \in R^2 | 1 < x^2 + y^2 < 5\} $ And $f:D \to R$ continuous s.t $f(2,0) = 2 $ and $ f(-2,0) = -1$ Show that ...
0
votes
1answer
22 views

show that F is a gradient vector

Denoting $x{\bf{i}}+y{\bf{j}}+z{\bf{k}}$ by ${\bf{r}},$ let ${\bf{F}}({\bf{r}})=\|{\bf{r}}\|^2\;{\bf{r}}$. Show that ${\bf{F}}$ is a gradient vector and find $f$ such that $\nabla f={\bf{F}}$. Any ...
1
vote
1answer
22 views

Continuity of a two-variable function?

I have to check if the function $$f(x,y)=\frac{x^3y}{2x^6+y^2}$$ can be defined at $f(0,0)$ so that it is continuous at $(0,0)$. I first checked for $\lim_{(x,y)\to(0,0)}\frac{x^3y}{2x^6+y^2}$ along ...
2
votes
2answers
60 views

Calculate limit with integral

Hi I have a problem with following limit: $$\lim_{x\rightarrow\infty}e^{-x}\int_{0}^{x}\int_{0}^{x}\frac{e^u-e^v} {u-v}\ \mathrm du\ \mathrm dv$$ as a hint i got that i should use de l'Hospital. So: ...
0
votes
0answers
32 views

The vector field with form $\textbf{w}=\lambda(r)\textbf{r}$

my question states: Suppose that the vector field $\textbf{w}$ has the form $\textbf{w}=\lambda(r)\textbf{r}$ where $\textbf{r}=x{\hat i}+y{\hat j}+z\hat k$ and $r=\sqrt{x^{2}+y^{2}+z^{2}}$. a) ...
0
votes
1answer
21 views

What are the partial derivatives of $f(x,y)=x^3-3xy^2$?

What are the partial derivatives of $f(x,y)=x^3-3xy^2$? I thought $f_x= 3x^2-3y^2$ and $f_y= -6y$ however I need to ensure whether these are correct.
0
votes
0answers
41 views

Double Integral over the region of an ellipse cut off by a circle

I've been stuck on this question for awhile. I need to calculate the double integral $\iint_R \frac{1}{r^3} dA$ using polar coordinates. R is the region displayed below: The ellipse has centre ...
0
votes
0answers
20 views

Flux integral of a 2D vector field.

Let $F(x,y) = (x^2, y^2)$ and $C$ be the upper half of the unit circle with positive orientation. Then find the flux integral of $F$ over $C$. The normal used here will be inward-facing normal. ...
0
votes
2answers
25 views

Soft Question: Taking Multivariable Calculus vs. Introduction to Proofs Class [closed]

Would taking multivariable calculus be boring in comparison to an introduction to proofs class for someone good at Calc. I and II? The multivariable calculus class would not cover topics like Green's ...
2
votes
0answers
39 views

Showing that $\|.\|$ is a norm of the space of 1-forms $\Omega^1(U)$, where $U\subset\mathbb{R}^n$.

Let $U\subset\mathbb{R}^n$ and let $\Omega^p(U)$ denote the vector space of $p$-forms ($p\in\mathbb{N}$). Define the isomorphism $\Phi:\Omega^{1}(U)\to\Omega^{n-1}(U)$ as ...
0
votes
0answers
31 views

Derivative of function of one variable with respect to function of two variables

I'm looking to find the derivative of a function of one variable with respect to a function of two variables: $$ \frac{df(x)}{dg(x,y)} $$ I'm not entirely sure whether this is possible in the first ...
1
vote
1answer
65 views

Stoke's Theorem Theory Question

If $C$ is the boundary of a surface $S$ and $\phi$ and $\psi$ are arbitrary smooth scalar fields, Show that $$\int_C \phi \nabla \psi \cdot dr = - \int_C \psi \nabla \phi \cdot d r = \iint_S (\nabla ...
1
vote
1answer
32 views

Proving that a function maintain a certain equation

Hey guys so here a new question, I need to prove that the function $$g(x,y,z)=f(\frac{1}{y}-\frac{1}{x},xye^{\frac{-z^2}{2}})$$ maintain the equation $$x^2g_x+y^2g_y=-\frac{x+y}{z}g_z$$ while ...
0
votes
2answers
34 views

How to find the shortest distance between $z^2 -xy = 1$ and the origin using Lagrange multiplier?

My task is this: Find the points on the surface $z^2 -xy = 1$ with the shortest distance to the origin. My work so far: Let $f(x,y,z) = x^2 + y^2 + z^2$ and $g(x,y,z)=z^2 -xy -1$ then we have to ...
0
votes
1answer
31 views

continuous of minimum

Let $\Omega$ be an closed bounded and connected domain in $R^n$ and $h(x,t)$ is continuous in $\Omega\times [0,T]$.Let $$ H(t)=\min_{x\in\Omega} h(x,t) $$ How to prove $H(t)$ is continuous ? What I ...
0
votes
1answer
16 views

Bounds of a solid region triple integral

I am trying to find the integral of (e^y)dV, where D is the solid region bounded by planes y=1, z=0, y=x, y=-x, and z=y. I set the integral up as (e^y)dzdydz but I am unsure how to determine the ...
0
votes
0answers
30 views

A calculation in polar coordinates in $\mathbb{R}^n$; pointwise gradient bounds

Suppose we have a smooth function $f : \mathbb{R}^n \to \mathbb{R}$, and it is given that $f(0) = 1$. It is also known that $|\nabla f| \leq \gamma |f|$ everywhere. The question is, what is the radius ...
0
votes
0answers
39 views

Volume of an elipsoid using Gauss' Divergence Theorem

Question: Let $F= (0,0,z)$ be a vector field. Use Gauss' Divergence Theorem to calculate the volume of the ellipsoid $x^2+y^2+2z^2=1$ My attempt: $$r(a,b) = ...
0
votes
1answer
16 views

Notation: $\sum_i \dfrac{\partial A_i}{\partial x_i} \boldsymbol{e}_i$ using $\nabla$.

I would like to write $\sum_i \dfrac{\partial A_i}{\partial x_i} \boldsymbol{e}_i$ using the $\nabla$ operator if possible, where $\boldsymbol{A}=A_1\boldsymbol{e}_1 + A_2\boldsymbol{e}_2 + ...
0
votes
0answers
15 views

Line integral on an arbitrary Path

Compute $∫_C −e^y\sin(x) dx + e^y\cos(x) dy + dz$, where $C$ is an arbitrary smooth path from $(0, 0, 1)$ to $(π, π, 0)$. Make sure to check you satisfy the hypotheses of any theorems you use. ...
0
votes
1answer
27 views

Compute the line integral where $C$ is an arbitrary smooth path from $(0,0,1)$ to $(\pi,\pi,0)$

Compute $∫_C −e^y\sin(x) dx + e^y\cos(x) dy + dz$, where $C$ is an arbitrary smooth path from $(0, 0, 1)$ to $(π, π, 0)$. Make sure to check you satisfy the hypotheses of any theorems you use. I went ...
0
votes
0answers
13 views

Find point or points where a tangent plane is parallel to the plane z=0

At which point (or points) on the ellipsoid $x^2 + 4y^2+z^2=9$ is the tangent plane parallel to the plane $z=0$ I am assuming that this occures at points $(0,0,-3)$ and $(0,0,3)$ because $z^2=9$ at ...
0
votes
1answer
22 views

Checking differentiability at a point of multivariable functions.

I am a bit confused on to how to prove differentiability in higher dimensions. My understanding is this so far: 1) If partial derivatives of a function exist and are continuous then it follows that ...
0
votes
1answer
21 views

Is a function which is bounded along a path bounded? [closed]

Does boundedness along a path of a multivariable function imply that the function is bounded?
1
vote
1answer
56 views

Maximum of function containing two variables $x$ and $y$

If $x+y+\sqrt{2x^2+2xy+3y^2} = k(\bf{Const.})\;,$ Then $\max(x^2y)\;,$ Where $x,y\geq 0$ $\bf{My\; Try::}$ Let $x^2y=z\;$ Then we get $$x+\frac{x^2}{z}+\sqrt{2x^2+\frac{2z}{x}+\frac{3z^2}{x^4}} = ...
0
votes
1answer
32 views

Solving the integration problem by use of fundamental theorem of calculus and chain rule

In one test the question said $$f(x,t)=\int_{0}^{g(x,t)} e^{-u^2} du$$ Now how I can calculate $\partial^2f/\partial t^2$ ? I have this idea: $$\int_{0}^{g(x,t)} e^{-u^2} du=F(g(x,t))-F(0)$$ ...
1
vote
1answer
11 views

Multivariate Non-Differentiability

This example says that "continuous partial derivatives imply differentiability but not vice-versa". Based on transposition logic, I would then assume that if a multivariate function has discontinuous ...
0
votes
1answer
18 views

If the integral over an area is zero is the integral of the gradient also zero?

Say I know that $\int\int v_z dx dy = 0$ over some area with $dA = dx dy$. $v_z$ is a function of $x$ that "points" in $z$. Is this enough to say that $\int\int \frac{\partial v_z}{\partial x} dx dy = ...
2
votes
1answer
33 views

$A\cup \text{Fr}A$ is a manifold

I'm dealing with the following exercise from Spivak's "Calculus on Manifolds": Let $A\subset \mathbb R^n$ be an open set such that the (topological) boundary of $A$, $\text{Fr}A$ is an $n-1$ ...
0
votes
0answers
11 views

Is this curl operator with surface normal and tangential components valid?

Is this curl operator valid? $\nabla \times \mathbf{A} = (\partial_{\tau_1} A_{\tau_2} - \partial_{\tau_2} A_{\tau_1}) \hat{\mathbf{n}} - (\partial_n A_{\tau_2} - \partial_{\tau_2} A_n) ...
0
votes
1answer
15 views

Calculating interesting volume (Also general multiple variable function equation)

First off, What does $f(x,y,z) = 1$ mean? versus $f(x,y,z) = x$. I have trouble with this as I am not sure choosing an initial $x$ factors into the function. Similarly with choosing a corresponding ...
-1
votes
1answer
26 views

Calculate the Area of Toroidal

i need help with the integral of this function. $ f(x,y)=\left( x-b \right) ^{2}+{y}^{2}={a}^{2}$ Note: $0 < a < b$ I know the graphics of this function is this ...
2
votes
0answers
69 views

If $\int_{y=0}^{y=b}\int_{x=0}^{x=a}f(x,y)dxdy=ab$, what is $f(x,y)$?

If $\int_{y=0}^{y=b}\int_{x=0}^{x=a}f(x,y)dxdy=ab$, for all $a,b$ between $0$ and $1$, what is $f(x,y)$? I know that the answer is 1. But I want to know how you would get there, if you didn't know ...
0
votes
0answers
8 views

Square-integrability in one variable of two-variables function

Let $h: [0,T]^2 \to \mathbb{R}_+$ be a measurable function. Assume that $$ \int_0^T\left(\int_0^T h(r,s)ds\right)^2dr < \infty .$$ Does this imply that $$ \int_0^T \left(h(r,s)\right)^2 dr < ...
0
votes
1answer
43 views

How to solve equations of the form $ax^4+bx^2+c$

Question: Find the values of $x$ for the following equation $$108x^4-507x^2+300$$ My attempt: I have the solution and it says to use the quadratic formula. However how would I apply that here? ...
0
votes
0answers
19 views

Double Integral with Residues

I'm trying to solve the integral $$\int_a^b\int_a^b\frac{dxdy}{1+\left(x^2+y^2\right)^\alpha}$$ where the constant $\alpha$ is real-valued and in the range $\alpha\in[1/2,\infty)$. The bounds $a$ ...
3
votes
2answers
31 views

Implicit derivation to find $\partial x/\partial v$?

I saw this question: $$\begin{cases} x^2+y^2=u \\ x\sin y+y=v\end{cases}$$ What is the $\partial x/\partial v$? I think it should be $1/\sin(y)$ because $\partial v/\partial x=\sin y$, but the ...
2
votes
0answers
18 views

Multiple vs single change of variable (in convolution)

Let $f$ and $g$ be in $L^1(\mathbb{R})$. Then $$\int_{\mathbb{R}^2}|f(y)g(x-y)|\, dy\, dx=\int_{\mathbb{R}^2}|f(y)| |g(t)|\, dy\, dt=\|f\|_1\|g\|_1$$ This calculation appears in proving that $f*g\in ...
0
votes
0answers
20 views

Prove that integration of a differential $k$-form is independent of choice of basis

This is Exercise 4 of Section 33 of Munkres' "Analysis on Manifolds" book: (Let $A$ be an open set in $\mathbb{R}^k$.) If $\eta$ is a $k$-form in $\mathbb{R}^k$ and if ...
2
votes
4answers
103 views

Finding the volume of the region bounded by $z=\sqrt{\frac{x^2}{4}+y^2}$and $x+4z=a$. Cylindrical coordinates.

I would like the answer to preferably be done using either using a surface integral, or an integral with substitutions. But anything other than this is alright, if nothing else exists. I have to find ...
1
vote
1answer
25 views

Unit base vectors in a new coordinate system

Let's assume we have a function $f:\Omega =R^2 \rightarrow R $ $f(x,y)=x+2xy+x^2y$. Obviously our unit base vectors on $\Omega$ are $e_x=\hat{i}$ and $e_y=\hat{j}$. Now we want to change the ...
0
votes
1answer
35 views

Question the convergence of $\iiint_{x^2+y^2+z^2\geq1}\frac{e^{\sin(x+y+z)}}{(x^2+y^2+z^2)^p}$ in dependence of $p$

Question the convergence of $$\iiint_{x^2+y^2+z^2\geq1}\frac{e^{\sin(x+y+z)}}{(x^2+y^2+z^2)^p}$$ in dependence of $p$. In class we did ...
0
votes
1answer
20 views

Find the volume of the region outside cone and inside sphere.

Find the volume of region outside the cone $\varphi = \frac{\pi}{4}$ and inside the sphere $\rho =4cos(\varphi)$. Solution Attempt: I can visualize the surfaces and see that the volume is two ...
1
vote
2answers
43 views

Proving no vector potential for gravitation field defined on all of $\mathbb{R}^3 -$ origin

Let: $$F=\frac{x,y,z}{(x^2+y^2+z^2)^{3/2}}$$ Show that there is no vector potential for F which is defined on all of $\mathbb{R}^3 - \text{origin}$ I can find a vector potential which is not ...
0
votes
1answer
24 views

Computing taylor series for two variables

Question: Compute all the terms in the taylor series for the following function around the point $(1,1)$ $$f(x,y) = x^2 + y^2$$ My attempt: $$f_x= 2x, \ f_y= 2y, \ f_{xx}= 2, \ f_{yy}=2$$ So we ...
0
votes
0answers
19 views

Solenoidal vector field

I am to prove (using the equations for gradient, divergence and curl in spherical polar coordinates) that vector field $\mathbf{w}=w_{\psi}(r,\theta)\hat e_{\psi}$ is solenoidal, find ...