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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1answer
23 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 ...
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1answer
37 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.- ...
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2answers
63 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 ...
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1answer
45 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
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2answers
60 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
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1answer
32 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 ...
2
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2answers
47 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
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1answer
68 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
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1answer
30 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 ...
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2answers
37 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 ...
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1answer
29 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 ...
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2answers
41 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 ...
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3answers
46 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_1^2 \int_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 circle on the x ...
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5answers
71 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
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1answer
40 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.
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1answer
23 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
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1answer
23 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
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2answers
58 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 $z_0$ ...
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1answer
68 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 ...
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2answers
36 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 ...
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1answer
20 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
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0answers
39 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 ...
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2answers
55 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? $x^ay^b=a^ab^b(a+b)^{-(a+b)}(x+y)^{a+b}$
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1answer
34 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. ...
2
votes
1answer
16 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 ...
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0answers
34 views

Limits in multivariate

Find four types of functions in two variables, where converting to polar coordinates seemingly tells you that the limit exists but the limit doesn’t actually exist. (By types, I mean they cannot be ...
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0answers
33 views

Lagrange Multiplier

Let $f$ be a function in two variables. To find the maximum and minimum over a closed contour $g = c$ (where $g$ itself is a function of two variables), we discussed how one could use the Lagrange ...
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3answers
25 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
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1answer
22 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 ...
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1answer
33 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 $(x_0,y_0)$ ...
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1answer
45 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 ...
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1answer
34 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 ...
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0answers
40 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
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1answer
42 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*} ...
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1answer
54 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 ...
0
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0answers
28 views

Lower boundary for hessian eigen values

This question appears on a differential geometry test i am trying to solve. Let $f$ be a function of $x,y$ and $k>0$ a positive real number. the function obeys the following: $f(0,0)=0$ and ...
2
votes
4answers
66 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
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2answers
143 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 ...
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1answer
40 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) = ...
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1answer
36 views

Let $f: \mathbb{R}^n \rightarrow \mathbb{R}^m $ be differentiable and $K \subset \mathbb{R}^n$ be compact and convex. Show $f$ is Lipschitz on $K$.

The Assignment: Let $f: \mathbb{R}^n \rightarrow \mathbb{R}^m $ be continuously partial differentiable and let $K \subset \mathbb{R}^n$ be compact and convex. Show $f$ is Lipschitz on $K$. A ...
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2answers
36 views

solving double integrals

I'm trying to solve a double integral: $\displaystyle \int_{0}^{\frac{1}{2}}\int_{0}^{\frac{1}{2}-y}24xy\; dx \;dy$ I first solved in respect to $y$, making the $x$ a constant and plugged in the $y$ ...
1
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1answer
27 views

Verification of Stokes Theorem

I want to verify Stokes Theorem for the surface $$ \Phi = \{ (x,y,z) \in \mathbb R^3 : z = x^2 - y^2, x^2 + y^2 \le 1 \} $$ and the vector field $F(x,y,z) := (y,z,x)$. For this I use the ...
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1answer
40 views

Solid of revolution [closed]

Part A) Let T be a region in the first quadrant of the xz-plane bounded by the curves x 4 = z, x = 2, z = 16. Using the methods of 1-variable calculus, calculate the volume of the solid obtained by ...
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0answers
21 views

set notation for directional derivative: what does this comma mean?

Under the proof, for L,M what does the M stand for? Another function/set or domain? Also under the proof section, can someone give it to me in layman's terms why the first inequality is less than or ...
0
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1answer
22 views

Application of Implicit Function theorem for this problem

Let $f: \mathbb{R}^3 \to \mathbb{R}^2$ be of class $C^1$; write $f$ in the form $f(x,y_1,y_2)$. Assume that $f(3,-1,2) = \mathbf{0}$ and $$ Df(3,-1,2) = \begin{pmatrix} 1 & 2 & 1 \\ 1 ...
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3answers
80 views

Find the directional derivative of the scalar field

Find the directional derivative of the scalar field: $f(x,y,z)=\log(x^2+y^2+z^2)$ at $P_0(1,1,1)$ in the direction of the straight line $\ P_0P $ where $P=(3,2,1)$ What I have done: ...
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0answers
18 views

Integral invariant under parametrization

Consider a continuous function $F(z,p)\colon \Omega\subset\mathbb{R}^N \times \mathbb{R}^N \to \mathbb{R}$ and the functional $$ \mathcal{F}(u)=\int_{a}^{b}{F(u(t),u'(t))\,dt}. $$ Prove that ...
1
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0answers
44 views

Integration in polar coordinates?

Given $$ A=\begin{pmatrix} a & b \\b & c \end{pmatrix}, x=(x_1,x_2), (Ax,x)>0 $$ and $$(x,y)=x_1\cdot y_1+x_2\cdot y_2$$ I'm trying to prove that $$ \int_{-\infty}^\infty ...
0
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0answers
38 views

Lower boundary on hessian eigen values [duplicate]

This question appears on a differential geometry test i am trying to solve. Let $f$ be a function of $x,y$ and $k>0$ a positive real number. the function obeys the following: $f(0,0)=0$ and ...
1
vote
4answers
37 views

Help calculating the surface area given by the polar curve: $r=2(1-\cos\theta)$

I want to calculate the surface area given by the curve: $$ r = 2(1-\cos(\theta)) $$ using an integral. I have thought about doing this: $$ x = r\cos(\theta), \, y = r\sin(\theta) $$ $$ \iint r \,dr ...