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
7 views

Taking derivative of energy of wave equation

Consider the variable coefficient, real valued wave equation $$ u_{tt} - \nabla \cdot (c^2 \nabla u) + qu = 0, \quad u(x,0) = \phi(x), \quad u_t(x, 0) = \phi(x), $$ where $c, q \geq 0$ depend only on ...
3
votes
1answer
44 views

Construction of a continuous function which maps some point in the interior of an open set to the boundary of the Range

I was studying the Inverse function theorem when I came across the following problems : (Let the closed set $V$ i.e the range have non-empty interior) Does there exist a continuous onto ...
1
vote
1answer
531 views

Absolute extrema

Find the absolute extrema of the function $f(x,y)=2xy-x-y$ over the region of the xy-plane bounded by the parabola $y=x^2$ and the line $y=4$ I was wondering if I needed to use Lagrange multipliers to ...
-2
votes
1answer
24 views

Infimum and supremum of two variable function [on hold]

How can I find the infimum and supremum in $\mathbb{R}^{2} $ of this function $$ f(x,y)=(2x^2+y^2-1)(x^2+y^2-1)+1 $$? Thanks EDIT: Forgive me if I did not add my thoughts but I did not know where to ...
2
votes
1answer
41 views

Definition of partial derivatives from Rudin's PMA

It's the definition of partial derivative from Rudin's PMA. Why he consider $(25)$ for real functions $f_i$? What about if $f_i$ in $(25)$ replaced by vector-valued function ...
0
votes
1answer
23 views

Function is differentiable in all the points of its domain

I need to proof that this function is differentiable in all the points of its domain. I know that this is true if the function is a function $\in C^k$ and a function is $C^k$ if is composition of ...
0
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0answers
8 views

Negative eigenvalue of a hessian matrix entails a local decrease in function value?

I was reading up on non-convex optimization, and I can across this sentence: "Since Hessian(f(w)) has a negative eigenvalue, there is always a point that is near w which has smaller function value" ...
-4
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0answers
29 views

How to prove the following questions by IBP? (Integrated By Parts) [on hold]

So this is the question that I have to solve. I know this is related to IBP, but Have no idea how to start and prove... need help
0
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1answer
42 views

Graph of $f(x,y) = \frac{3x^2 y}{x^2+y^2}$ near the origin

I am trying to graph the function $f : (x,y) \mapsto \frac{3x^2 y}{x^2+y^2}$ on a TI-89 Titanium. I have noticed that no matter how many times I zoom in toward the origin the graph appears identical. ...
1
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1answer
33 views
+50

Complicated surface integral/line integral.

Problem Compute the integrals $$I=\iint_\Sigma \nabla\times\mathbf F\cdot d\,\bf\Sigma$$ And $$J=\oint_{\partial\Sigma}\mathbf F\cdot d\bf r$$ For $F=(x^2y,3x^3z,yz^3)$, and ...
0
votes
2answers
44 views

Method of characteristics - finding the particular solution using initial conditions

I am trying to use the method from my previous question to solve this PDE: $$ 3u_x + 2u_t = \cos x $$ with initial condition $u(x,0) = x^2$. So I need to solve these: \begin{align} \frac{dx}{ds} ...
1
vote
0answers
15 views

singular $1$ cube - Boundary of $2$ chain

This is an exercise from "Calculus on Manifolds" by Michel Spivack (first edition, p.100): If $c$ is a singular $1$-cube in $\mathbb{R}^2-\{0\}$, with $c(0)=c(1)$, show that there is an integer ...
3
votes
1answer
173 views
+50

Biharmonic Equation in a Rectangle with Some Uncommon Boundary Conditions

Consider the following boundary value problem (BVP) $$\matrix{ {{\Delta ^2}H = 0,} \hfill & {} \hfill & {{\rm{in}}\,} \hfill & \Omega \hfill \cr {\partial _y^2H = 0} \hfill & ...
3
votes
0answers
87 views
+50

A singular $n-$cube and a circumference defined the border than 2-cube

This is an exercise from "Calculus on Manifolds" by Michel Spivack (first edition, p.100): If $c$ is a singular $1$-cube in $\mathbb{R}^2-\{0\}$, with $c(0)=c(1)$, show that there is an integer ...
0
votes
1answer
25 views

Finding correct variation for $\rho$ in spherical coordinate integration

I am having some trouble and looking for help on calculating the moment of inertia about the z axis of the region bound by the cone $z=\sqrt{3(x^2+y^2)}$ and the sphere $x^2+y^2+z^2=a^2$ if the ...
2
votes
1answer
2k views

Does the Divergence Theorem Work on a Surface?

The divergence theorem in $\mathbb{R}^3$ says that the integral of the divergence of a vector field over a solid $\Omega$ in $\mathbb{R}^3$ equals the flux through the surface of $\Omega$ denoted by ...
0
votes
1answer
23 views

Differentiating a function composition

Given $g:R^n \rightarrow R^k$ and $h:R^k \rightarrow R$, we have $f(x) = h(g(x))$. Using the chain rule, we can differentiate $f(x)$ to get $f'(x) = \nabla^Th(g(x))g'(x)$ My question is why do we ...
4
votes
1answer
67 views

Integral of an unbounded function as a solution of $\nabla^2\boldsymbol{A}=-\boldsymbol{J}$

While studying the equivalence between the Biot-Savart and Ampère's laws I have only found proofs of the fact that$$\boldsymbol{A}(\boldsymbol{x})=\frac{\mu_0}{4\pi}\int_V ...
2
votes
3answers
59 views

Why doesn't the limit $\lim_{(x,y) \rightarrow (0,0)} \frac{ e^{x+y} - x - y}{\sqrt{x^2 + y^2}}$ exist?

Why is this limit non-existant? $\lim_{(x,y) \rightarrow (0,0)} \frac{ e^{x+y} - x - y}{\sqrt{x^2 + y^2}}$ I can't seem to find $2$ different paths that would show it is non-existant.
12
votes
1answer
240 views

Mathematical meaning of certain integrals in physics

While studying on texts of physics I notice that differentiation under the integral sign is usually introduced without any comment on the conditions permitting to do so. In that case, I take care of ...
1
vote
1answer
19 views

Computing Gauss's of a sphere

The vector field given as $\vec{F}=\frac{\left \langle x,y,z \right \rangle}{\sqrt{x^{2}+y^{2}+z^{2}}}$ The region $D=\left \{ a^{2}\leq x^{2}+y^{2}+z^{2}\leq b^{2} \right \}$ I've some ...
0
votes
3answers
31 views

Intersection of 2 planes - find vector of intersection

I was trying to figure out the curve of intersection of these 2 planes: $$3x - y + z = 4 $$ $$ y + z = 2.$$ I realize it will be a straight and not curved, and feel like I should be able to do the ...
0
votes
1answer
24 views

How to find centroid of this region bounded by surfaces

I am having difficulty find the centroid of the region that is bound by the surfaces $x^2+y^2+z^2-2az=0$ and $3x^2+3y^2-z^2=0$ (lying above $xy$ plane, consider the inner region). I know the first ...
1
vote
1answer
28 views

“vector” vs “point” in definition of directional derivative

Given a function $f\colon \mathbb R^n\to\mathbb R$, and given $x,v\in\mathbb R^n$, it is customary to define the "directional derivative of $f$ in the direction $v$ at the point $x$" by $$ D_v f(x) = ...
2
votes
3answers
51 views

Evaluate this integral using cylindrical coordinates

Find the volume of the solid bounded above by the paraboloid of revolution $z^{2}=x^{2}+y^{2}$ And below by the $xy$ plane, and on the sides by the cylinder $x^{2}+y^{2}=2ax$ We take $a>0$. ...
3
votes
1answer
40 views

How does Green's theorem apply here?

Let $D$ be the region delimited by $$\partial D: \begin{cases} C_1: x^2 + y^2 = 5^2\\ C_2:(x-2)^2+y^2= 1\\ C_3:(x+2)^2+y^2 = 1\\ C_4: x^2+(y-2)^2= 1\\ C_5: x^2+(y+2)^2= 1 \end{cases} $$ I've sketched ...
3
votes
2answers
278 views

Implicit Function Theorem

Q: Use Implicit Function Theorem to show that there exists a unique solution of the equation $x^{e^y} + y^{e^x} = 0$ in a neighborhood of the point $(0, 0)$. I tried to satisfy three conditions of ...
0
votes
1answer
31 views

Evaluate $\int_{0}^{1}\int_{x}^{1} y^2 \sin(2\pi \frac{x}{y})dydx$

I am trying to evaluate this integral: $$\int_{0}^{1}\int_{x}^{1} y^2 \sin(2\pi \frac{x}{y})dydx$$ $$=\int_{0}^{1}\int_{0}^{1} \chi_{[x,1]}(y) y^2 \sin(2\pi \frac{x}{y})dydx$$ ...
0
votes
0answers
24 views

Change of variables for path integral.

Let $G=C^\infty([0,1];\mathbb{R}^d)$ be smooth paths, then for the path $A\in G$, consider the translation operator from $G$ to itself $T_A:G\to G$ $$T_A(g)(t):=g(t)+A(t).$$ Does there exist a ...
5
votes
2answers
407 views

Second derivative of a vector field

I wonder how to treat the "second derivative" of a vector field. For example, imagine we have a vector field $f:\mathbb{R}^n \rightarrow \mathbb{R}^n$. Then we evaluate the derivative at two points ...
1
vote
1answer
13 views

When can a function have its variables seperated

Suppose I have a function $f(x,y,z)$. I need to know when one can write it as $$f(x,y,z)=a(x)\cdot b(y) \cdot c(z)$$ where $a, b, c$ are functions. I don't want to know what they are, but just whether ...
0
votes
0answers
34 views

evaluate this region using gauss's theorem (only using the triple integral 'part')

Evaluate $$\iiint _{D}\vec{\nabla} \cdot\vec{F}\,dV$$ with $$\vec{F}=\left \langle x^{2},y,z \right \rangle$$ $$D=\left \{ \left ( x,y,z \right )|x^{2}+y^{2}+1\leq z\leq 5 \right \}$$ ...
1
vote
1answer
74 views

Laplace Equation in a Cylinder with Some Uncommon Boundary Conditions

While I was working on some theorems in PDEs, I encountered the following axisymmetric boundary value problem $$\matrix{ {{\nabla ^2}H = 0} \hfill & {{\rm{in}}} \hfill & \Omega \hfill ...
1
vote
0answers
14 views

Generalize Implicit Differentiation to find Tangent Plane

For a function $F(x,y,z)$ with $(a,b,c)$ on the level surface $F(x,y,z)=k$, where $F(x,y,z)=k$ defines $z$ implicitly as a function of $x$ and $y$. Using the chain rule, assuming $F_z(a,b,c)\neq0$ ...
0
votes
1answer
15 views

Vector Valued Functions: Parametrize the intersection of 2 surfaces w/ trigonometric functions

The question asks: Parametrize the intersection of the surfaces using trigonometric functions. $$y^2-z^2=x-6$$ $$y^2+z^2=81$$ $\mathbf{r}(t)=$ ____ My first step was recognizing ...
0
votes
0answers
31 views

The derivative of a function of multiple variables

I am trying to understand a step in the theory section of my differential equations textbook. The author writes, For example, suppose we transform the first order differential equation ...
0
votes
0answers
17 views

Can an iterated integral over a box R ={(x,y,z)|x∈[0,a], y∈[0,b], z∈[0,c]} be expressed in eight different ways?

this is my first time on stack exchange so sorry if I am not following any guidelines. I received this exact question on a midterm and answered yes, it is possible, which was considered wrong on the ...
8
votes
1answer
2k views

Singular jacobian matrix?

I have a series of questions, in various degrees of befuddled muddledness (and they are related to my previous questions: this and this) First question: how do I do a change of variable if the ...
3
votes
1answer
119 views

Differentiability of a 2-variable function

In this morning's Mathematical Analysis 2 exam, students were asked to study the continuity and differentiability of: $$f(x,y)=\left\{\begin{array}{cc} ...
0
votes
1answer
34 views

Differentiating $- \sum_{n \in \mathbb{Z}^2} e^{i n \cdot \alpha}\int_0^E\frac{1}{4\pi t}\exp({\omega^2 t - \frac{|x - n - y|^2}{4t^2}})dt$ wrt $x$?

I have a formula for the Ewald method which can be used to speed up computations when working with periodic Green's functions. I will need to take the derivative of the function $G(x, y)$ with respect ...
0
votes
1answer
21 views

Chain rule in partial derivatives

I've come across the following expression in my textbook about the chain rule in partial differentiation that I don't quite follow To be more specific, it's the diferentiation of (6.9) right at the ...
2
votes
1answer
34 views

Showing that a function is injective

I am trying to show that the following function is injective in some neighborhood of $(0, 0)$: $f:\mathbb R^2 \rightarrow \mathbb R^2$ given by $$f(x, y)=(\sin(x^3)\cosh(y), \cos(x^3)\sinh(y))$$ I ...
1
vote
1answer
39 views

Double integration over function with absolute values

I have having difficulty in how to solve the following double integral problem involving absolute values and the assumption that $\alpha > 1$: $\iint_{-\infty}^{+\infty} \frac{1}{1+|x|^\alpha} ...
1
vote
1answer
39 views

Knowing a scalar field from its Laplacian and gradient

As a part of the "rabbit hole" I am descending in order to understand the meaning of the integrals of a not-so-rarely found derivation of Ampère's law, I am trying to understand how to see the ...
0
votes
1answer
42 views

Extending Picard-Lindelöf (quick proof check)

I am trying to show that, if $F:\;[0,1]\times \mathbb{R}^{n}$ is uniformly Lipschitz continuous in the $\mathbb{R}^{n}$ variables, the system: $$f^{(n)}(t)=F\big(t,\,f(t),\,\dots,\, ...
0
votes
0answers
14 views

To show the inverse operator Inv is continuous

The book said we can use the identity $X^{-1}-Y^{-1}=X^{-1}•(Y-X)•Y^{-1}$ to prove the Inverse operator Inv is $C^0$. Also, assume Inv is $C^{(r-1)}$, how can I prove it is $C^r$ and $C^{(r+1)}$? ...
3
votes
1answer
213 views

Is zero vector potential for Helmholtz decomposition of curl and divergence free vector fields necessary?

Helmholtz's theorem tells us that a sufficiently smooth vector field $\mathbf{A}$ can be decomposed into curl and gradient free parts as the gradient of a scalar potential plus the curl of a vector ...
3
votes
1answer
19 views

Show that the vector field $\vec F=(xf(u),xg(u))$ is not conservative

I'm trying to prove that the vector field $\vec F=(xf(u),xg(u))$ with $u=xy$ is not conservative. I suppose that there is a function $\phi$ so that $\nabla \phi= \vec F$. So I need to satisfy that: ...
0
votes
0answers
26 views

why the concepts of partial derivatives and differentiability need a open set?

In order to define the limit of a function $f: A \subset \mathbb{R^n} \to \mathbb{R^m}$ at a point $x_0 \in \mathbb{R^n}$ we need $x_0$ to be a limit point of $A$. But in order to speak about partial ...