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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0answers
109 views

Simplifying a Vector Integral

While reading the book - Theory and Applications of Boltzmann Transport Equation by Cercignani (I am not a math student), I found this integral which I am unable to understand. Note that $\xi_i , ...
2
votes
1answer
173 views

Show that $L[v^2] := \Delta(v^2) + \frac{2}{w} \sum_{i=1}^n w_{x_i} (v^2)_{x_i} \ge 0$

Let $u$ be a function such that $$ \Delta u + \lambda u = 0 $$ for some $\lambda \in \mathbb R$, also let $w$ be a function such that $$ \Delta w + \beta w < 0. $$ for some $\beta \in \mathbb R$. ...
0
votes
2answers
33 views

Question on derivation of vector identites and using some symbolic manipulations

Let $f,g : \mathbb R^n \to \mathbb R$, then for the gradient we have the product rule $$ \nabla(fg) = (\nabla f) \cdot g + f \cdot (\nabla g). $$ And by $\Delta(f) = \mbox{div}(\nabla(f)) = \nabla ...
3
votes
2answers
190 views

Find maximum of $P$

Let $$P = \frac{{{x^2}}}{{{x^2} + yz + x + 1}} + \frac{{y + z}}{{x + y + z + 1}} - \frac{{1 + yz}}{9}.$$ Find maximum of $P$ where $x, y,z$ are nonnegative real numbers such that ${x^2} + {y^2} + ...
0
votes
1answer
82 views

evaluating surface integral with divergence theory

If I have to calculate the surface integral of $\iint_S A \cdot n\ \mathrm {ds}$ where $A= 3zi-2xj+5x^2zk$ and $S$ is the surface of the cylinder $x^2+y^2=4$ and lying between $z=0$ and $z=4$ in the ...
2
votes
1answer
78 views

Divergence Theorem To Calculate Surface Integral

$M=${$(x,y,z):x^2+y^2+z^2=16$,$\sqrt {x^2+y^2}\leq z$} We are asked to find the surface area of this surface. This is my way: $\partial M=${$(x,y,z):x^2+y^2+z^2=16$,$\sqrt {x^2+y^2}= z$} so the ...
0
votes
1answer
46 views

Arc Length with Vector-Valued Functions, Part B

Consider the path of a particle in a conservative force field represented by the vector-valued function $$r(t)= \left(4(\sin t−t \cos t), 4( \sin t+t \sin t), \frac{3}{2} t^2 \right).$$ A) Find the ...
4
votes
2answers
504 views

Why absolute values of Jacobians in change of variables for multiple integrals but not single integrals?

If $g:[a,b]\to\mathbf R$ is a change of 1D coordinates, then the formula is: $$ \int_{g(a)}^{g(b)}\,f(x)\,dx = \int_a^b\,f(g(t))\frac{dx}{dt}\,dt. \qquad\text{(1)}$$ If $T=\{x=f(u,v); y=g(u,v)\}$ ...
2
votes
0answers
123 views

Computation of the Frenet-Serret trihedron in $\Bbb L^3$ (Lorentz-Minkowski space)

Consider $\Bbb L^3 = (\Bbb R^3, \langle , \rangle)$, with the convention $$\langle (x_1,y_1,z_1), (x_2,y_2,z_2)\rangle = x_1x_2+y_1y_2 - z_1z_2$$ and $\| v \| = \sqrt{|\langle v, v \rangle|}$. Let ...
0
votes
0answers
91 views

Finding maximum rate of change of total derivatives

consider $PV =nRT , P,V,T =$ pressure , volume , temperature respectively. $nR =$ constant let $n=R=1$ differentiate with respect to $t$ (time) $dP/dt = ∂P/∂T * dT/dt + ∂P/∂V * ...
2
votes
1answer
85 views

Determining Lipschitz constant for a special vector field

Let us be given a vector field $v: C \subset \mathbb R^n \to \mathbb R^n$ that has the special structure given by $$ v(x) = \alpha(x) \begin{pmatrix} x_1 \\ \vdots \\ x_n \end{pmatrix} $$ with a ...
0
votes
2answers
40 views

Study limit of a two variable function for every point of domain

Let $$f:\mathbb{R}^2\rightarrow \mathbb{R}/f(x,y)=y^2-2x^2y+6x^3-3xy+2y-6x$$ I need to show that: $\lim\limits_{(x,y)\to(x_0,y_0)}\dfrac{f(x,y)}{y-3x}$ exists. So: ...
0
votes
1answer
43 views

Surface integral using Stokes' theorem

$$ \int_\Gamma y\,dx+z\,dy+x\,dz $$ when $\Gamma$ $= \{ (x,y,z): x^2+y^2+z^2=9\}$ $\cap$ $x+y+z=0$ There's a theorem that states: $\int_S(\nabla \times \vec F)\cdot d \vec S$= $\int_S(\nabla \times ...
0
votes
2answers
26 views

Double integral integration limits

For multivariable function $ f(x,y) = xy^2$ how to find the interval of integration, the question gives those inequalities $ x \le 2, y \ge 0, y \le x $.
1
vote
1answer
50 views

What is meant by the continuity of the Hessian matrix

I have a simple and short question: "What is meant by the continuity of the Hessian matrix?" I guess it means that all the second partial derivatives of a function $f$ are continuous functions? is ...
0
votes
1answer
44 views

Implicit function theorem- a misunderstanding?

I have the following claim in my notebook: If $F(x,y)=0$, and $\frac{\partial F}{\partial y} \neq 0 $ and $\frac{\partial F}{\partial x} \neq 0 $ then: $F$ is not continuously differentiable (i.e.- ...
1
vote
2answers
81 views

Showing a two-variable function is continuous

The problem asks to show that $$f(x,y) = \left\{ \begin{align} \frac{x^3y^2}{x^4+y^4}, & (x,y) \neq (0,0), \\ 0, & (x,y) = (0,0), \end{align} \right.$$ is continuous at the origin, however it ...
1
vote
1answer
55 views

Is an integral without a differential component on a finite number of points just a sum?

Is an integral $$\int_{\lbrace 1, 2, 3 \rbrace} f(x)$$ simply the sum $$\sum_{x=1,2,3} f(x)?$$ I ask this question because of the generalization to multiple dimensions of integration by parts ...
2
votes
2answers
80 views

Integrating $ \int_0^2 \int_0^ \sqrt{1-(x-1)^2} \frac{x+y}{x^2+y^2} dy\,dx$ in polar coordinates

I'm having a problem integrating $ \displaystyle\int_0^2 \int_0^ \sqrt{1-(x-1)^2} \frac{x+y}{x^2+y^2} \,dy\,dx$. I drew the graph, and it looks like half a circle on top of the $x$ axis. I tried ...
1
vote
1answer
96 views

Surface area of lateral section of paraboloid

Here is my 2D parabola curve. The x,y locations of each point of interest are displayed, and the equation of the parabola and the line are given as well. I want to create a hollow 3D Paraboloid by ...
1
vote
2answers
184 views

Surface Integral of a Right Circular Cone

Use a surface integral to show that the surface area of a right circular cone of radius $R$ and height $h$ is $\pi R \sqrt{h^2+R^2}$. Hint -- Use the parametrization $x=r\cos\theta$, $y=r\sin\theta$, ...
1
vote
1answer
71 views

When is $x\mapsto |x|^{s-1}x$ a diffeomorphism?

Consider the function $f:B^n\rightarrow B^n$ from the disk to itself $$f(x)=\vert x\vert^{s-1}x$$ where $s>0$ and we are considering the euclidean norm (we define the function to be $0$ in the ...
3
votes
1answer
40 views

Divergence Theorem Question

$$\iint\limits_\sum f \ d \sigma = \iiint\limits_S \operatorname{div} \textbf{f} \ dV$$ $$\operatorname{div} \textbf{f}=1+2+3=6$$ After this, we could multiply $6$ by the volume of the sphere ...
0
votes
2answers
46 views

Double Integration: $\iint_D\ e^{30x}\ dA$

I am having trouble with this double integral. I know I must set it up to have the $y$ values go from $x$ to $x+1$ and the $x$ values from $0$ to $1$. When I solved the integral I got the answer ...
0
votes
1answer
128 views

Finding volume between plane and paraboloid

Find the volume between bounded by $z=4$ and $z=x^2+y^2$.(Answer: $8\pi$) I thouhg using dievergence theorm ($\iint_KdivFdxdydz=\iint_SF\cdot\hat{n}dS$) for $\vec{F}=\big(\frac x 2,\frac y ...
0
votes
2answers
87 views

Integrating $\iint_R \sin(9x^2+4y^2)\ dA$

The question I'm trying to solve is: $\displaystyle \iint_R \sin(9x^2+4y^2)dA$, where $R$ is the region in the first quadrant bounded by $9x^2+4y^2=1$. I'm a little confused in solving this. Does ...
1
vote
3answers
395 views

Integrating $\int_1^2 \int_0^ \sqrt{2x-x^2} \frac{1}{((x^2+y^2)^2} dydx $ in polar coordinates

I'm having a problem converting $\int\limits_1^2 \int\limits_0^ \sqrt{2x-x^2} \frac{1}{(x^2+y^2)^2} dy dx $ to polar coordinates. I drew the graph using my calculator, which looked like half a ...
5
votes
5answers
224 views

Interpreting higher order differentials

I'm trying to understand Taylor's Theorem for functions of $n$ variables, but all this higher dimensionality is causing me trouble. One of my problems is understanding the higher order differentials. ...
0
votes
1answer
60 views

Solve an integral with Stokes' theorem

For $\gamma=\{(x,y,z):x^2+y^2+z^2=9,x+y+z=0\}$ with positive orientation, find using Stokes' theorem: $$ \int_\gamma ydx+zdy+xdz $$ Thanks.
0
votes
1answer
50 views

Integral with sum of differentials

Somebody can help telling what this integral means? $$\int (x+y) dx+dy$$ On the path $g(t)=(t,t^2)$ between 0 and 1
0
votes
1answer
27 views

Help with vectorial analysis problem

Let $\psi : \mathbb{R}^n \to \mathbb{R}^n$ and $f: \mathbb{R}^n \to \mathbb{R}$ be continuously differentiable functions. Let $X \in \mathbb{R}^n$, $Y=\psi(X)$ and $g=f \circ \psi$. Show that $Z ...
3
votes
2answers
108 views

If $f'(z_0)\neq 0$ then $f$ has an holomorphic inverse.

Problem: Let $U\subset\mathbb{C}$ be an open set, $f:U\to\mathbb{C}$ an holomorphic function of class $C^1$ and $z_0\in U$. Prove that if $f'(z_0)\neq 0$ then there exists a neighborhood $V$ of $z_0$ ...
2
votes
1answer
100 views

How to define integration over the boundary of a curve?

When learning about Stokes' theorem ($\int_{\partial \Omega} \omega=\int_{\Omega} \mathrm d \omega$), we are told that it is just a generalization of the 2nd Fundamental Theorem of Calculus $(\int_a^b ...
3
votes
2answers
60 views

Variable substitution in second order PDE

Consider the the PDE $$A(x, y)\partial_{xx} u + B(x, y)\partial_{xy}u + C(x, y)\partial_{yy}u=h(x, y) $$ Now I want to make a variable substitution $\xi=f(x, y), \eta=g(x,y)$, so I can get $u$ as a ...
0
votes
1answer
22 views

Showing $\int_{-1}^1\int_{-1}^1(u_x^2+2u_y^2+u^2-x^2y^2u)\, dx\, dy\geq c$.

Prove that for some $c\in\mathbb{R}$: $$G(u) =\int_{-1}^1\int_{-1}^1(u_x^2+2u_y^2+u^2-x^2y^2u)\, dx\, dy\geq c$$ for every $u \in H_0^1$. I know that $$G(u) ...
2
votes
0answers
44 views

How “separable” (not in that sense) is a polynomial?

Since "separable" is used for different meaning in separable polynomial and separation of variable, I am having trouble searching for anything related to my question. So I hope someone can help with ...
0
votes
2answers
83 views

explain this confusing algebraic identity?

Can anyone show, step-by-step, how the expression on the LHS can be turned into the expression on the RHS? $x^ay^b=a^ab^b(a+b)^{-(a+b)}(x+y)^{a+b}$
2
votes
1answer
85 views

How to prove that $F(x,y)=(f(x)h(y),g(y))$ is a diffeomorphism?

Let $F:\mathbb{R}^2\to\mathbb{R}^2$ be given by $F(x,y)=(f(x)h(y),g(y))$, where $h:\mathbb{R}\to\mathbb{R}$ is a diferentiable function and $f,g:\mathbb{R}\to\mathbb{R}$ are diffeomorphisms. ...
3
votes
1answer
2k views

Definition of cluster point

I'm studying if the book Multidimensional Real Analysis by Duistermaat and the definition of cluster point is: A point $a \in \mathbb{R}^n$ is said to be a cluster point of a subset $A$ if for every ...
0
votes
3answers
53 views

Find function $F(x,y)$ given its total differential

If I have a differential $dF = 2xy e^{xy^2} dy + y^2e^{xy^2} dx $, what are the steps to find the original function? Is there a formula?
0
votes
1answer
37 views

Evaluate derivatives y'(0),z'(0),y''(0),z''(0) of implicit functions y(x) and z(x)

Evaluate derivatives $y'(0)$, $z'(0)$, $y''(0)$ ,$z''(0)$ of implicit functions $y(x)$ and $z(x)$, where $y(0)=-1$ and $z(0)=1$, given by system of equations: $x+y+z=0$ and $x^2+y^2+z^2=0$ First ...
1
vote
1answer
53 views

Sufficient conditions for differentiability of multivariate functions

Claim: If a function $f:\mathbb R^2\to\mathbb R$ has partial derivatives in a neighborhood $D$ of $(x_0,y_0)$, and if these are continuous at $(x_0,y_0)$, then $f$ is differentiable at $(x_0,y_0)$ ...
4
votes
1answer
65 views

How to show that $\varphi(x,y)=(x+f(y),f(x)+y)$ is bijective?

Let $f:\mathbb{R}\to\mathbb{R}$ be a $C^1$ function such that $|f'(t)|\leq k<1$ for all $t\in \mathbb{R}$. Let $\varphi:\mathbb{R}^2\to\mathbb{R}^2$ be the function given by ...
1
vote
1answer
38 views

a question about multivariable integral!

If $\lfloor x \rfloor$ denotes the greatest integer in $x$, evaluate the integral$$ \iint_{R} \lfloor x+y \rfloor ~ \mathrm{d}x~ \mathrm{d}y$$where $R= \{(x,y)| 1\leq x\leq 3, 2\leq y\leq 5\}$. This ...
1
vote
0answers
59 views

Difficult Surface Integral

I am trying to perform a surface integral over kind of a weird shape. So the radius of the shape should be equal to the multiple of $3$ constants (one for each of the $x, y$ and $z$ directions) each ...
0
votes
1answer
226 views

Evaluate integral $\int\int xe^{xy} dx dy$, strange result after rearranging

I have to compute the following integral $$ \int_{-1}^0 \int_0^1 x\cdot e^{xy} dx dy $$ It exists according to WolframAlpha. Now I want to evaluate it, let $\varepsilon > 0$, then \begin{align*} ...
1
vote
1answer
79 views

Calculate area enclosed by curve

Calculate the area of the bounded surface enclosed by the curve $(x+y)^4 = x^2y$ with the help of the coordinate transformation $x = r\cos^2 t, y = r\sin^2 t$. As I see it the area is unbounded, so ...
2
votes
4answers
90 views

$f$ is one to one on the set $A$ consisting of all $(x,y)$ with $x>0$. What is the set $f(A)$

Let $f:\mathbb R^2\to\mathbb R^2$ be defined by the equation $$f(x,y)=(x^2-y^2,2xy).$$ Show that $f$ is one to one on the set $A$ consisting of all $(x,y)$ with $x>0$. What is the set ...
3
votes
2answers
286 views

Maximizing Area of Triangle in Circle

I was playing around with another example that I made up where I am trying to maximize the area of a triangle inscribed in a circle of radius. I want to do the problem using the method of Lagrange ...
0
votes
1answer
306 views

Prove that These Families of Level Curves are Orthogonal

From p. 79 in Brown's and Churchill's "Complex Variable and Application": Let the function $f(z) = u(x, y)+iv(x, y)$ be analytic in a domain $D$, and consider the family of level curves $u(x, y) = ...