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).

learn more… | top users | synonyms (1)

2
votes
2answers
184 views

A limit two variables

How can I compute or prove that $\displaystyle\lim_{(x,y)\to(0,0)}\dfrac{\mathrm{e}^{xy}-1}{\sqrt{x^2+y^2}}=0$?
0
votes
1answer
56 views

Find the volume inside

Find the volume inside the torus $\rho=\sin\phi$. First of all how can $\rho=\sin\phi$ represent a torus? I can't even visualise that. All Ideas are welcome, this looks like a 'food for thought ...
0
votes
1answer
42 views

Find $\int_0^1 \int_{3x}^3 (x^2+y^2)\sqrt{9-y^2}\hspace{1mm}dy dx$ [closed]

You can use a calculator after simplification if its not possible by hand All Ideas will be appreciated Also If you could find $$\int_0^1 \int_{3x}^3 x(x^2+y^2)\sqrt{9-y^2}\hspace{1mm}dy dx$$ ...
1
vote
1answer
37 views

Improper integral $\int_{B}\frac {1}{|x|^\alpha}dV$

Let B be the ball $|x|\le 1$, $x\in R^n$. For what $\alpha$ does $$\int_{B}\frac {1}{|x|^\alpha}dV$$ exists? I find it hard when it comes to generalize this statement in $R^n$. I've been able to do ...
2
votes
2answers
33 views

Maximum and minimum of $z=\frac{1+x-y}{\sqrt{1+x^2+y^2}}$

Find the maximum and minimum of the function $$z=\frac{1+x-y}{\sqrt{1+x^2+y^2}}$$ I have calculated $\frac{\partial z}{\partial x}=\frac{1+y^2+xy-x}{(1+x^2+y^2)^{\frac{3}{2}}}$ $\frac{\partial ...
4
votes
0answers
52 views

Improper multivariable integrals

I'm having trouble with the integral $$\iiint_{1\le x^2+y^2+z^2 }\frac{\mathrm{d}x~\mathrm{d}y~\mathrm{d}z}{xyz}$$ this is what I've done so far: $$\lim_{b\to +\infty}\int_1^b \int_0^{2\pi} ...
1
vote
1answer
26 views

Proving that a field satisfies stokes theorem.

The field is the classic $$F (x,y,z) = \left( \frac{-y}{x²+y²}, \frac{x}{x²+y²},0\right)$$ And the surface is the space between $x² + y² =1$ and $x+y=1$ at $z=0$ Since $ \nabla \times F = 0 $ and ...
0
votes
1answer
23 views

multivariable calculus double integration volume question

Use a double integral to find the volume of the solid bounded by graphs of the equations given by: $$\begin{align}z=xy^2, \text{ where: } &z>0\\&x>0\\&5x<y<2\end{align}$$ My ...
2
votes
3answers
76 views

Find $\iiint_E (1-x^2-2y^2-3z^2)~\mathrm{d}V$, where $E$ is the region inside the ellipsoid $x^2+2y^2+3z^2=1$ [closed]

My textbook asked to use a calculator to find this. Not sure how to setup the triple Integral.
6
votes
3answers
208 views

find $ \int_0^4\int_0^4\int_0^4 \sqrt{x^2+y^2+z^2}\,dx\,dy\,dz$

I am looking for an approximation to the nearest integer of $$ \int_0^4\int_0^4\int_0^4 \sqrt{x^2+y^2+z^2}\,dx\,dy\,dz.$$ Wolfram alpha gives up and says "computation time exceeded". I tried, ...
1
vote
0answers
28 views

Evaluating a path integral without being given a parametrization

Find the mass of the wire formed by the intersection of the sphere $x^2 + y^2 + z^2 = 1$ and the plane $x + z = 0$ if the density of the wire is $6y^2$ grams per unit length. I am completely stuck on ...
0
votes
2answers
49 views

Calculations of double integral of $xy^2$ over the region between $y=0$ and $y=4-x^2$

Find $\iint_D f~\mathrm{d}A $ where $f(x,y) = xy^2$. Region $D$ is between $y=0$ and $y=4-x^2$ If you draw the graph, you can see that it equal to $0$, however if you calculate the answer ...
1
vote
3answers
82 views

interpreting triple integrals

The limits of integration on a double integral are justified because I'm essentially finding the area underneath two functions. Namely, first I integrate (say, with respect to dx) and I get an area ...
1
vote
0answers
42 views

Are the two properties of a function equivalent?

$f(x)$ is a function defined on $\Bbb R^n$. $A$: $\forall x,y$ $$ |f(y)-f(x)-\nabla f(x)^T(y-x)| \le \frac{\beta}{2}\|y-x\|_2^2 $$ $B$: $\forall x,y$ $$ \| \nabla f(y)-\nabla f(x)\|_2 \le \beta ...
2
votes
2answers
64 views

Why does this vector derivation hold?

I have the following variables/matrices: $$A \in \mathbb{R}^{m \times n} , \quad p \in \mathbb{R}^{n}, \quad \Sigma \in \mathbb{R}^{m \times m}, \quad w \in \mathbb{R}^{m}$$ where $\Sigma$ is a ...
1
vote
2answers
49 views

Do projections on $\mathbb{R}^2$ transform straight lines to straight lines?

A linear transformation $P:\mathbb{R}^{2} \longrightarrow \mathbb{R}^{2}$ is called projection if $P \circ P =P$. The question is: If $P$ is a projection then $P$ transforms straight lines ...
0
votes
0answers
18 views

Help with calculating line integrals and potential functions [duplicate]

May you please help me with this questions? 1) Among all smooth, simple closed curves in the plan, oriented counterclockwise, find one along which the work done by the following vector is greatest: ...
-1
votes
0answers
37 views

potential functions and line integrals

May you please help me with this questions? 1) Among all smooth, simple closed curves in the plan, oriented counterclockwise, find one along which the work done by the following vector is greatest: ...
0
votes
1answer
19 views

Multivariable Calculus Application Question: Utility and MCRS

If a student has a utility function given by $$U(x_1, x_2) = −x_1 + > 10x_ 2^2 − 2x_1x_2$$ where $X_1 = 5$ and $X_2 = 20$. If this student was to eat $5$ less hot meals per month, estimate the ...
1
vote
1answer
75 views

line integral…

Calculate $$\int_Γ f \, d\ell$$ for $f(x,y) = y, \; y=x^{1/2}$, $ x $ is in $[2,6]$. I know (now) that it means that: $$\int_\Gamma f \, d\ell=\int_a^b f(\Gamma(t)) \cdot \|\dot\Gamma(t)\| \, dt$$ ...
1
vote
1answer
21 views

Find and classify the critical points of the function $f(x,y) = x^3 +2y^3 - 3x^2 -24y + 6$

I have to find and classify the critical points of the function: $$f(x,y) = x^3 +2y^3 - 3x^2 -24y + 6$$ I have said that $$f_x = 3x^2 -6x=0 $$ $$3x(x-2)=0$$ $$x=0, 2$$ $$f_y=6y^2-24=0$$ $$y=±2$$ I ...
2
votes
1answer
72 views

How can I find the point (X, Y, Z) which minimizes this quantity?

I have a number of equally powerful light sources $L_i, 1 \le i \le N$ at points within a cube $(x_i, y_i, z_i), -1 \le x_i, y_i, z_i \le 1$. The intensity of each light falls off with distance ...
-1
votes
0answers
32 views

Constraint optimization with lagrangian

I am having trouble regarding the general steps one needs to take in order to solve an constraint optimization using Lagrangian. More specifically, I want to maximize objective equation $f(x,y,z,w)$ ...
0
votes
1answer
49 views

Proof of the rank theorem in Rudin's PMA book

I am studying Rudin's proof of the rank theorem (theorem 9.32 in Principles of Mathematical Analysis.) We have an invertible function $H(x)$ defined on an open set. He claims we can "shrink" the open ...
2
votes
0answers
38 views

The 3rd and 4th Critical Point?

I must find and classify all the critical points in the following function: $$ f(x,y)= x^2 + y^2 +x^2y +4$$ I have said that $$f_x=2x+2xy=0$$ $$ 2x = -2xy$$ $$ \frac{ 2x}{\ -2x}=y $$ $$y=-1$$ $$f_x = ...
0
votes
1answer
40 views

Multivariable Calculus application

A firm is producing cylindrical containers to contain a given volume. Suppose that the top and bottom are made of a material that is $N$ times as expensive as the material used for the side of ...
0
votes
0answers
19 views

Simplification of an integral comprising of vector-variables

How can I evaluate the simplify the integral $\int \rho (\bf{r^{\prime}})\, \delta (\frac{\sigma}{2}-r) d \bf{r^{\prime}}$ where $\delta$ is the dirac-delta function given that $\rho$ is constant ...
0
votes
2answers
38 views

Two-dimensional Taylor linearisation

I have to perform a first order taylor expansion of a function $f(\vec{x}) = f(x+u,y+1)$ at the point $\vec{a} =(x,y)$. My solution reads $$ f(\vec{x}) \approx f(x,y) + \left( \begin{matrix} ...
0
votes
2answers
29 views

How to treat Dirac delta function of two variable?

We can treat one variable delta function as $$\delta(f(x)) = \sum_i\frac{1}{|\frac{df}{dx}|_{x=x_i}} \delta(x-x_i).$$ Then how do we treat two variable delta function, such as $\delta(f(x,y))$? ...
0
votes
0answers
27 views

Area of the portion of the cylinder $x^2+y^2 = 9$ for which $-1 \leq z \leq 2$ and $ 0 \leq \theta \leq \pi/2$

Problem: Find the area of the portion of the cylinder $x^2+y^2 = 9$, for which $-1 \leq z \leq 2$ and $ 0 \leq \theta \leq \pi/2$ I first solved this by parametrizing the surface. $x = 3\cos(u)$ , ...
4
votes
1answer
77 views

Why generalize the derivative for multivariable functions? [duplicate]

Sorry if this is a dupe (did a search, couldn't find anything). In single variable calculus, if the following limit exists: $$\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h},$$ then this expression ...
4
votes
0answers
57 views

Differential forms and determinants

2-forms are defined as $du^{j} \wedge du^{k}(v,w) = v^{j}w^{k}-v^{k}w^{j} = \begin{vmatrix} du^{j}(v) & du^{j}(w) \\ du^{k}(v) & du^{k}(w) \end{vmatrix}$ But what if I have two concret ...
0
votes
0answers
31 views

Partial Derivative - Almost there!

I must prove that $$\ t \frac{\partial Z}{\partial T} - s \frac{\partial Z}{\partial S} = (x-1) \frac{\partial Z}{\partial X}$$ In this question $$ z = x^2 + 3xy^2 $$ $$x= 1-st^2 $$ $$y=st$$ ...
1
vote
2answers
47 views

What is the potential function of the field $\left(\frac{-y}{x^2+y^2},\frac{x}{x^2+y^2}\right)$

The vector field is obviously conservative on every closed domain that doesn't encompass the point $(0,0)$, so there must be a potential function. I've got $\arctan(\frac{x}{y})$ for $x$ unequal to ...
0
votes
0answers
13 views

What order should I evaluate divergence and coordinate transformation if I want to use a different coordinate system?

I have a vector field in Cartesian coordinates. I need to find its divergence, but I need the divergence to be in spherical coordinates. However, the divergence of this field is far easier to ...
0
votes
1answer
24 views

Trigonometric Partial Derivative

I need to find $$\frac{\partial Z}{\partial U} \text{ and } \frac{\partial Z}{\partial V}$$ for a $z=f(x,y) = \cos(xy) + y\cos(x)$. After a bit of an internet search, I think I have found the ...
0
votes
1answer
33 views

Partial Derivative Stickler.

I am having trouble with a question with partial derivatives. Here is the question: Let $\rho = \sqrt{x^2 + y^2 + z^2}$ Show that $$\frac{\partial ^2\rho}{\partial x^2} + \frac{\partial ...
0
votes
0answers
20 views

Theorem 4.6 in Spivak's Calculus on Manifolds

Could you elaborate on the proof please? This is how I would prove the theorem: Since $\Lambda^n(V)$ is $1$-dimensional, $\omega=\alpha(\phi_1\wedge\phi_2\wedge...\wedge\phi_n)$ for some $\alpha$ ...
0
votes
2answers
24 views

Having a bit of trouble with min/max distance from sphere to point

The sphere is $x^2 + y^2 + z^2 = 81$ and the point is $(5,6,9)$ I am using Langrane multipliers , but the answers I am getting are so far off. I will post my system of equations soon. I found ...
1
vote
1answer
29 views

Vectors with given angle and magnitude

Give an example of vectors $\mathbf{v}$ and $\mathbf{w}$ such that the angle between $\mathbf{v}$ and $\mathbf{w}$ is $\frac{2\pi}{3}$ and $\|\mathbf{v} \text{ x } \mathbf{w}\|=\sqrt{3}$. Should I ...
0
votes
0answers
36 views

Paraboloid Curvature calculation methods

If we have a paraboloid generated as a surface of revolution of the 2d function $f(x)=ax^2+b$, the equation of the 3d graph is $f(x,y)=ax^2 + ay^2+b$. The gaussian curvature of a 3d graph $f(x,y)$ is ...
3
votes
2answers
93 views

Solving a particular system of differential equations

The problem I'm trying to solve is this: $X'(t) \in \mathbb{R}^3 \,, \, \omega = (\omega_1,\omega_2,\omega_3) $ Find the general solution for $$X'(t) = \omega \times X(t)$$ After doing the cross ...
1
vote
1answer
23 views

Cobb Douglas Difficulty

Show that the Cobb-Douglas production function, for Labour costs L and Capital costs K, $P(L, K) = AL^{\alpha}K^{1-\alpha}$ satisfies the equation: $$L\frac{\partial P}{\partial L} + ...
1
vote
1answer
26 views

Need Help Understanding How To Integrate With An Implicit Variable

My calculus is really rusty (damn Mathematica/Matlab) and I was wondering if anyone could help me with an equation I am having trouble integrating. I have attached a snapshot of the paper I am trying ...
0
votes
0answers
32 views

A question regarding calculus in several variables.

Let $f:\Bbb{R^n}\to \Bbb{R}$ be a continuous function. I refer to this proof of the Mean Value Theorem for several variables. Here $g(t)=f(a+t(b-a))$. Hence ...
2
votes
1answer
27 views

Taking Fourier transform of integral-differential equation

If $u$ is a solution of the equation $$\frac{\partial}{\partial t} u(x,t) + \int_{-\infty}^{\infty} \text{sinc}(x-y) \cdot \frac{\partial^{2}}{\partial y^{2}} u(y,t) \ dy = 0,$$ with initial condition ...
0
votes
1answer
37 views

Transform square region to triangular region

How do you express x and y in terms of u and v so that the region $\{(u,v): 0\le u, v\le 1\}$ is mapped to the triangular region in the $xy$-plane with vertices $(0,0)$, $(1,0)$, and $(0,1)$? Now, ...
5
votes
0answers
39 views

Reference for the fact that a smooth function analytic on every line is itself analytic

Let $f \in \mathcal C^\infty(\mathbb R^p)$ ($p \geq 2$) be a smooth function such that the functions $g_d(t) := f(td)$ are all analytic for all $t \in \mathbb R$ and all $d \in \mathbb R^p.$ (i.e. $f$ ...
2
votes
1answer
30 views

Explanation of Lagrange Equation with Chain Rule

I am just reading through some lecture notes explaining the Lagrange Equation, and I am a bit confused with some chain rule stuff, I get to the part with: $$\frac{\partial F}{\partial y} = ...
0
votes
1answer
28 views

How to calculate the partial derivatives of the composition $F(u(s,t),v(s,t))$?

Could someone help me to understand how to do this problem? I believe Partial Derivatives are used. Thanks!