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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8
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2answers
83 views

Help to understand changing order of integration

I have a problem I have been working on, with the solution but the thing is I don't really understand how it is done. The question, is to compute, $$\int_0^1 \int_{9x^2}^9 x^3\sin(8y^3) \,dy\,dx $$ ...
0
votes
3answers
46 views

$\mathbb{R}^k$ is of measure zero in $\mathbb{R}^l$, $k < l$. [closed]

How do I show that $\mathbb{R}^k$ is of measure zero in $\mathbb{R}^l$, with $k < l$?
0
votes
1answer
31 views

using the divergence theorem

Use Gauss Divergence Theorem to compute $$\int \int \limits_S F\cdot n \,dS$$ where $n$ is the outward normal for the following: $S$ is the surface $x^2 +y^2=z^2$ and $z \in [0,1]$, and ...
0
votes
0answers
17 views

Conditions for always positive gradient of heat field in evolutionary thermo-elastic system

I am investigating stability and convergence of series of approximations for coupled thermoelasticity problem yielded by one-step recurrent time-integration scheme. I've managed to show that the ...
0
votes
1answer
14 views

differential of a product of two quantities

So far I've always blindly done a sort of product rule on a differential like this $$ d(\rho V) = \rho dV + V d\rho$$ I'm now wondering if it is also legitimate to write $$ d(\rho V) = \rho dV + V ...
0
votes
0answers
18 views

Dam Break Problem

Is $u=0$, $h=1$ at every point where $x<-t$? $\frac{dx}{dt}=-1$ on $C_-$ and $\frac{dx}{dt}=1$ on $C_+$ only initially (at $t=0$) so how do we know that characterisitics intersect at every ...
0
votes
1answer
64 views

partial derivative of $\arctan ( \frac{x+y}{1+xy})$

partial derivative of $\arctan ( \frac{x+y}{1+xy})$ I am lost in finding the partial derivatives of the function. I started with the formula $\frac{1}{(1+x^2)}$. But it gets really complicated. Is ...
3
votes
3answers
91 views

The limit of $xy/(y-x^3)$ at $(0,0)$ does not exist

How to prove that $$\lim_{(x,y)\to (0,0)}\frac{xy}{y-x^3}$$ doesn't exist? Obviously, if $f(x,y)=\frac{xy}{y-x^3}$ and, for instance, $\gamma_1(t)=(t,0)$, we have $\lim_{t\to0}f(\gamma_1(t))=0$. We ...
0
votes
2answers
16 views

Finding limits of a volume triple integral in cylindrical coordinates

Find the volume between the cone $z=\sqrt{x^2+y^2}$ and the sphere $x^2+y^2+z^2=1$ that lies in the first octant (i.e., $x>0$, $y>0$, $z>0$) using cylindrical coordinates. It is obvious that ...
1
vote
2answers
25 views

Finding the critical points of $f(x,y) = x y^2 - x^2 y + x y$

Trying to find the critical points of $f(x,y) = y^2x - yx^2 + xy$. I took partial derivative with respect to x, so $F_x = y^2 - 2xy + y$ $F_x = y(y - 2x + 1)$ Then with respect to y, $F_y = 2xy - ...
0
votes
0answers
16 views

Implicit solution to Method of Characterstics

If I have that, $\sqrt cu+v=$ constant along the characteristic lines $x+\sqrt c t=$ constant Where $u=u(x,t), v=v(x,t)$ and $c$ is constant. Why is it that $\sqrt cu+v=f(x+\sqrt ct)$ where $f$ is ...
1
vote
1answer
34 views

The method of Lagrange's Multipliers

I used the method of Lagrange's multpliers to find the maximum of $f(x,y,z)=\ln x+\ln y+3\ln z$ on the portion of the sphere $g(x,y,z)=x^2+y^2+z^2=5r^2 \ ; r>0$ where $x>0, y>0, z>0$ . I ...
0
votes
0answers
6 views

Taking divergence of a field?

Given $J = (z-hut)J_osin[B(\frac {Δz}{2} - |z|)]$. I want to find $∇.J$, its confusing because I don't really see any r, theta, or psi directions. I only see z-direction, which isn't r because r is a ...
0
votes
1answer
8 views

the diagonal angle between a theta and phi vector and the x axis as well as the derivative

Find w, the angle ∠rox. That is, the angle from the red vector r to the positive ...
1
vote
1answer
24 views

limits of r in cylindrical coordinates

Find the volume of a sphere $x^2+y^2+z^2\leq 1$ contained between planes $z=1/2$ and $z=1/\sqrt2$ using cylindrical coordinates. So the limits of $\theta$ would be $0$ to $2\pi$. Limits of $z$ would ...
1
vote
1answer
17 views

Two Definitions of Critical Points

Let $f:\mathbb{R}^n\to\mathbb{R}^m$ be given such that it is continuously differentiable. According to Wikipedia, a "critical point" of $f$ is a point $p\in\mathbb{R}^n$ such that: 1) According to ...
3
votes
1answer
30 views

Is the isosurface of a smooth function a smooth surface?

suppose $ f(x)\in C_c^{\infty}(R^n, R)$, an infinitely differentiable function with compact support, from $R^n$ to $R$. If $f\not\equiv 0$, is the boundary of its support, i.e. $\partial\{x\in R^n: ...
0
votes
0answers
18 views

Proof Check: For every partition $P$ of box $B$: $ \frac {F_+(B)}{V(B)} \leq \min_{c \in P}\frac {F_+(C)}{V(C)}$

I have to prove that for every partition $P$ of box $B$: $$\min_{c \in P}\frac {F_-(C)}{V(C)} \leq \frac {F_-(B)}{V(B)} \leq \frac {F_+(B)}{V(B)} \leq \max_{c \in P}\frac {F_+(C)}{V(C)}$$ Where V is ...
1
vote
0answers
26 views

the second derivative for a composite function

Let $f: U\rightarrow \mathbb{R}$ where $U\subseteq X$ a normed vector space and suppose $f(\mathbf{u})\in C^{2}$.Let $\mathbf{x}\in U$ and let $r>0$ be such that $B(\mathbf{x},r)\subseteq U. $ ...
8
votes
3answers
77 views

Why is Green's theorem asymmetric in $x$ and $y$?

Green's theorem is $$\oint_{\partial D} (P\, dx+Q\, dy) = \iint_D dx\,dy \: \left ( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right)$$ where one can see that the RHS is ...
0
votes
2answers
41 views

How do I see that a linear function $L: \mathbb R^n \rightarrow \mathbb R^m$ is smooth?

How do I see that a linear function $L: \mathbb R^n \rightarrow \mathbb R^m$ is smooth ? I see that $L$ is indeed differentiable with $(DL)_x = L$ by definition of the derivative. But how do I ...
-2
votes
1answer
40 views

Differentiate intergral function

I found this equation when i read article (Cooperation among Competitors: Some Economics of Payment Card Associations by Jean Tirole), $$\int ^ \infty _x (y+const)dF(y)$$ where F(y) is CDF. he ...
1
vote
2answers
46 views

Volume of a solid between sphere and cylinder

Find the volume of a solid inside a sphere $x^{2} + y^{2} + z^{2} = 4$ and the cylinder $x^{2} + y^{2} + 2y = 0$ Any help is appreciated
0
votes
2answers
27 views

Finding area of a region

Let S be the part of plane $x+2y-z+5=0$ that lies inside cylinder $y^{2}+z^{2}-2y+4z-4=0$ Find the surface area of S I know that I have to define a function that describes S (in terms of x,y,z) and ...
1
vote
0answers
13 views

How to Determine the Direction of Green's Theorem (Work)?

Use Green's Theorem to calculate the work done by the given force field $\vec{F}$ in moving a particle counterclockwise once around the indicated curve $C$. $\vec{F} = 5x^2y^3\vec{i} + 7x^3y^2 ...
2
votes
0answers
46 views

Prove there exists a infinitely differentiable function whose value of partial derivatives of all orders at $0$ is a given function

Let $n \geq 1$ be an integer, and $C: \mathbb{N}^n \to \mathbb{R}$ be a function. Prove that there exits a infinitely differentiable function $f: \mathbb{R}^n \to \mathbb{R}$ whose value and partial ...
0
votes
2answers
37 views

Green's Theorem Problem

Use Green's Theorem to calculate the integral $\int_CP\,dx+Q\,dy$. $P = xy, Q = x^2, C$ is the first quadrant loop of the graph $r = \sin2\theta.$ So, Green's Theorem takes the form $\int\int_D ...
2
votes
1answer
20 views

Cone into cartesian coordinates

Let $K$ be the surface of an infinite cone with circular cross section, vertex at the origin and axis lying along the positive $z$-axis. If the angle between the $z$-axis and the surface of the cone is ...
2
votes
0answers
42 views

Simple Vector Calculus Integral

A little stuck on what I assume is probably a fairly basic vector calculus integration problem, but I haven't been able to find any resources online that deal with multivariable integration in a way ...
2
votes
0answers
13 views

Please check my calculations involving directional derivatives and gradient vectors

Here is the question: If g is a function differentiable at $(a, b)$ such that $\nabla g(a, b) = (2, 3)$, then find all the vectors $\vec{u} = (x, y)$ with $x^2 + y^2 = 1 $ such that $D_\vec{u} g(a, ...
0
votes
0answers
24 views

Derive the equation of first variation for a flow of a vector field.

This is a problem from Susan Colley's Vector Calculus. I have trouble understanding the solution to it. Problem: Derive the equation of first variation for a flow of a vector field. That is, if ...
1
vote
0answers
31 views

In which space does the Lie derivative of a vector field really live?

I'm slightly puzzled about the space in which the Lie derivatives of vector fields live. I get the worrying impression that it lives in the double tangent bundle, which I know is not true, so i need ...
0
votes
1answer
35 views

How to find $∂^2z/∂x^2$

If we have $∂z/∂x=∂z/∂u+∂z/∂v$, what is the technique to find $∂^2z/∂x^2$? It seems like you just kind of square both sides but that seems like an informal way to think about it..
2
votes
0answers
22 views

Characteristics function and moments of multivariates

I have been reading this paper recently-- http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.199.2157&rep=rep1&type=pdf This is a paper by Nengjiu Ju who uses talyor series to express ...
1
vote
0answers
17 views

Lagrangian, Kinetic & Potential energy with two masses connected to three springs [migrated]

Two masses $m_1$ and $m_2$ are on a frictionless surface. They are connected by three springs with constants $k_1,k_2,k_3$. $k_1$ and $k_3$ are attached to walls and $k_2$ is between the masses. $k_1$ ...
-1
votes
1answer
39 views

Proof that gradient of $\det(A)$ with respect to $A$ is $\det(A) A^{-1}$ [closed]

How to prove that $\dfrac{\partial |A|}{\partial A} = |A|A^{-1}$ where $|A|$ is $\det(A)$ and $A$ is symmetric matrix?
0
votes
1answer
12 views

Divergence of a vector tensor product / outer product:

I'm currently studying the derivation of the RANS (Reynolds Averaged Navier Stokes) equations, used in the study of turbulence, and I've stumbled upon a step wich I don't understand very well. The ...
1
vote
2answers
52 views

Area in R2 bounded by 4 curves

What's the area between: $$y = 5x$$ $$y = 15x$$ $$xy = 8$$ $$xy = 4$$ ? I calculated all the points of intersection and got a double integral that looks something like this: ...
2
votes
0answers
33 views

How do I see that $U \cap f^{-1}(V)$ is open in $\mathbb R^n$, where $V \subset \mathbb R^m$ is an open neighbourhood of $f(x_0)$?

Suppose $U \subset \mathbb R^n$ is open, let $x_o \in U$, and let $f: U \rightarrow \mathbb R^m$ be differentiable in $x_0 \in U$. Let $V \subset \mathbb R^m$ be an open neighbourhood of $f(x_0)$. ...
0
votes
1answer
19 views

How Do I Parameterize This Line Segment?

I'm being asked to evaluate a line integral along a path C. C is the path from $(-1,2,-2)$ to $(1,5,2)$ that consists of 3 line segments parallel to the axes; first the $z$ axis, then $x$, then $y$. ...
0
votes
0answers
11 views

Show that $T(\partial \mathbb{H^n} \cap U)=\partial \mathbb{H^n} \cap V$ where T is a $C^1$ diffeomorphism

Let $U$ and $V$ be open subsets of $\mathbb{H^n}$ and $T: U \to V$ be a $C^1$ Diffeomorphism. Show that $T(\partial \mathbb{H^n} \cap U)=\partial \mathbb{H^n} \cap V$. where $\mathbb{H^n}=\{x \in ...
0
votes
0answers
16 views

What is the relation between the gradient and Jacobian?

Suppose I have a function $V(\vec r_1, ..., \vec r_N)$ My texts claims that: $$(\nabla_i V)^T = \frac{\partial V}{\partial \vec r_i}$$ Can someone confirm this relationship?
0
votes
0answers
37 views

Negative Answer to Triple Integral?

I am solving a triple integral problem for volume and I am receiving the correct answer, but my sign is wrong. Should this be a cause of concern, or is it normal to have to take the absolute value of ...
0
votes
1answer
21 views

Line Integral Problem - Theoretically Should Be Simple

Evaluate $\int_C P(x,y)dx + \int_CQ(x,y)dy$ where $P(x,y) = y^2, Q(x,y) = x$, C is the part of the graph $x=y^3$ from $(-1,-1)$ to $(1,1)$. So, I begin by parametrizing: $x=t, y=t^3$, which makes my ...
0
votes
2answers
28 views

Using Lagrange for finding Marshallian Demand

I want to find the marshallian demand function for the user function $u(x_1,x_2) = x_1^ax_2^{1-a}$ where $a \in (0,1)$. This is what I have so far: $$L = x_1^ax_2^{1-a} - \lambda(p_1x_1 + p_2x_2 - ...
1
vote
0answers
7 views

In the envelope theorem, why can I write my inputs $x$ and $y$ as a function of $\xi$?

This is a question about the envelope theorem. Suppose I have a maximization problem $$\max_{x,y} f(x,y,\xi)$$ such that $$g(x,y,\xi) \leq c$$ where $x$ and $y$ are control variables and $\xi$ is a ...
0
votes
1answer
16 views

Surface Area Double Integral Problem

Find the area of the portion of the plane $z = x + 3y$ that lies inside the elliptical cylinder with equation $\frac{x^2}{4}+\frac{y^2}{9} = 1.$ The formula for finding this is $\int \int_{R} ...
1
vote
1answer
88 views
+50

The path of the shock

Here I am using the shock speed to work out the path the shock takes. I don't understand why we cannot take the value of $u_{-}$ at $t=1/u_0$ i.e $u_{-}=u_0$. and calculate the speed of the ...
3
votes
0answers
34 views

Lagrange Multipliers Dilemma

In the problem $f(x,y) = xy$ and $g(x, y) = x^2 + 9y^2 = 18$ I get $y = 2λx$, $x = 18λy$ and $x^2 + 9y^2 = 18$ (the constraint). All is fine, but I feel like I'll get two different answers ...
0
votes
0answers
21 views

Find the Volume Contained Inside a Sphere and Cylinder

Find the volume of the region that lies inside the cylinder $x^2 + y^2 = 1$ and $x^2 + y^2 + z^2 = 4.$ I've attempted to break this down into 2 sections: a pure cylinder, which continues until it ...