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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4answers
48 views

Area of circle (double integral and cartesian coordinates)?

I know that the area of a circle, $x^2+y^2=a^2$, in cylindrical coordinates is $$ \int\limits_{0}^{2\pi} \int\limits_{0}^{a} r \, dr \, d\theta = \pi a^2 $$ But how can find the same result with a ...
1
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0answers
21 views

Comprehension question about derivative in one point

Find the derivative of $f$ in $(x_{0} , y_{0})^{T}$ for: $$f(x,y)=\binom{x^4+2x^2y^2+y^4}{x^4+2x^2y^2+y^4}$$ Is it right to derivate $\partial x$ and $\partial y$ with $(x_0,y_0)^T$ ...
2
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0answers
33 views

Using Line Integrals to solve Amperes Law

I have been reading the forums on how to solve the integral form of ampere's law and I have worked out that the correct way to solve it is to get rid of the dot product by realizing that |B ∙ dr| is ...
0
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1answer
23 views

Prove this property of the Hessian

I have been reading about the hessian for a scholar work about optimization and I find this property: Let be $H_{P_0}$ the determinant of the hessian matrix for the Lagrangian function $\mathscr{L}(x,...
1
vote
1answer
47 views

the volume of the solid limited by the surfaces $x^2+z^2=4$ and $y^2+z^2=4$.

I want to find the volume of the solid limited by the surfaces $x^2+z^2=4$ and $y^2+z^2=4$. I have broken the volume into sixteen pieces, one of which can be represented by the following section: $0\...
0
votes
2answers
34 views

Integral of the level curve $ye^{2x}$

$\int_{0}^{2}\int_{0}^{4} ye^{2x}$ $dydx$ The book says the answer is $32(e^4-1)$ I had to do u-substitution when doing it, maybe that's where I went wrong. First integrating with respect to y, ...
-1
votes
1answer
58 views

How to prove this version of the fundamental theorem of calculus for curves in the closure of a domain

Dear Downvoters: if you leave a comment, you can influence the way this post gets modified, if you don't this post might never satisfy you - even though I keep editing Let $\Omega \subseteq \mathbb{...
3
votes
1answer
57 views

Find the limit of $\lim\limits_{(x,y)\to (0,0)} \frac{x^2y^2}{x^2y^2+(x-y)^2}$

Find the limit of $$\lim\limits_{(x,y)\to (0,0)} \frac{x^2y^2}{x^2y^2+(x-y)^2}$$ So, I know that $$\lim\limits_{x \to x_0} f(x)=c \Leftrightarrow \forall (x_n)\subseteq D\setminus\{x_0\}, x_n\to x_0:...
1
vote
2answers
40 views

Different notation for position vectors? Domain/Range?

What is the difference between this notation for position vectors? Are there any differences in domain and range? $$ \mathbf{r}=x{\mathbf{\hat{e}}_x}+y{\mathbf{\hat{e}}_y}+z{\mathbf{\hat{e}}_z} \qquad ...
3
votes
1answer
47 views

Show the triple integral given is equivalent to $\frac{15\pi}{16}$

Evaluate $$\iiint_E\;z \, dV$$ where E is enclosed between the spheres $x^2 + y^2 + z^2 = 1$and$x^2 + y^2 + z^2 = 4$ in the first octant. I'll be honest. My first ...
3
votes
0answers
26 views

$f$ is uniformly continuous only if $g$ is constant

Let $g:\mathbb R\to\mathbb R$ be continuous and define $f:\mathbb R^2\to\mathbb R$ by $f(x_1,x_2)=g(x_1x_2)$. Show that $f$ is uniformly continuous only if $g$ is a constant function. I'm not sure ...
0
votes
0answers
28 views

Double Integration in Polar Coordinates

$\iint 2x-y$ $dA$ in the first quadrant and enclosed by $x=0$ $y=x$ and $x^2+y^2=4$ Since the function is enclosed in the first quadrant then $0 \leq \theta \leq \frac{\pi}{2}$ and since $y=x$ and $x=...
0
votes
1answer
22 views

Find the vector v that has norm equal to 3 and has the same direction as the vector <0,1,-1>

What I did was normalized v, which gives $v=\sqrt{0^{2}+1^{2}+-1^{2}}$ then I divided that by the norm of the vector with the same direction so that $u=\sqrt{2}/3$ and multiply that by vector v's ...
0
votes
2answers
63 views

Does this double integral require u substitution?

$\int_0^1\int_0^3 e^{x+3y}$$dxdy$ $3y$ is treated like a constant so I am really just dealing with x here. How do I go about integrating this?
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1answer
11 views

Jordan measure of a convergent sequence

Suppose we have a convergent sequence a_n , where n goes from 1 to infinity. Why is the Jordan measure of this sequence equal to zero?
2
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0answers
36 views

Minimize a huge two-variable logarithmic-trigonometric-radical expression (MSU entrance early July 2016)

Minimize \begin{align}R(a,x)&=\sqrt{13+\log_a\left(\cos\left(\frac xa\right)\right)^2+\log_a\left(\cos\left(\frac xa\right)^4\right)}+\sqrt{97+\log_a\left(\sin\left(\frac xa\right)\right)^2-\...
1
vote
1answer
61 views

Taylor expansion $f(x)=f(0)$

The following taylor expansion of the function $f(x)$, requires $f(x)$ to have a derivative up to what order? $$ f(x)=f(0)+f'(0)x+f''(0)x^2/2+\mathcal{O}(x^3)$$ My solution: Based on the Taylor'...
2
votes
4answers
67 views

Find the maximum of $U (x,y) = x^\alpha y^\beta$ subject to $I = px + qy$

Let be $U (x,y) = x^\alpha y^\beta$. Find the maximum of the function $U(x,y)$ subject to the equality constraint $I = px + qy$. I have tried to use the Lagrangian function to find the solution for ...
-1
votes
1answer
14 views

Partial integration for smooth functions with compact support

Let $d\in\mathbb N$ $\lambda$ be the Lebesgue measure on $\mathbb R^d$ $\Omega\subseteq\mathbb R^d$ be open Why can we use partial integration to obtain $$\int_\Omega\phi\Delta\psi\;{\rm d}\...
-1
votes
2answers
21 views

differentiability for trigonometric function in two variable

Given $$f(x,y)=\left\{ \begin{matrix} 0 & (x,y)=(0,0)\\ \frac {\sin\left(x^2-xy \right)}{\vert x \vert} & (x,y) \neq (0,0) \end{matrix} \right.$$ Proved continuity, I have to seek for ...
0
votes
0answers
33 views

Multidimensional change of variables for pdf integration

I have a very simple question, but I could not find the answer, so I have to ask this here: Given is a multidimensional pdf $f(x_1, ..., x_n)$. $x_1, ..., x_n$ are Carthesian coordinates. We want to ...
2
votes
4answers
71 views

Prove that $\int_0^1\frac{x^y-1}{\log x}\mathrm dx=\log(1+y)$

The title says it all - I currently can't find a good way to start. Tried rewriting it into a line integral, but I really don't see a way to solve this right now. I'd appreciate any hints.
0
votes
2answers
9 views

Given $\angle (\vec{u} , \vec{v}) = 30^\circ$, $\Vert \vec{w} \Vert = 4$ and $\vec{w}$ is $\perp$ to both find $\vec{u} \cdot \vec{v} \times \vec{w}$

I am given the following problem: Knowing that the angle between the unit vectors $\angle (\vec{u} , \vec{v}) = 30^\circ$, $\Vert \vec{w} \Vert = 4$ and that $\vec{w}$ is orthogonal to both of ...
2
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0answers
34 views

Zorich's Mathematical Analysis, Volume II

Springer just published a new English version of Vladmir Zorich's two-volume Mathematical Analysis. I was looking at the second volume. It seems to have sections on both Multivariable/Vector Calculus ...
1
vote
3answers
152 views

Is there any metric $d$ on $\mathbb R$ and $a \in \mathbb R$ such that the function $f:\mathbb R \to \mathbb R$, $f(x)=d(x,a)$ is differentiable?

Let $d$ be any metric on $\mathbb R$ , then I know that the two variable scalar field $f: \mathbb R^2 \to \mathbb R$ , $g(x,y)=d(x,y)$ is never differentiable . Now what I want to ask is this : Let $a ...
1
vote
2answers
74 views

Is there a number that can be represented as $A^B=B^A$ after $16$ , where A and B are 2 distinct real numbers? [duplicate]

The number $16$ can be represented as $2^4=4^2$. Is there a (real) number after $16$ that can be represented as $A^B=B^A$? Where A and B are two distinct real numbers.I've checked many numbers and I'm ...
2
votes
1answer
19 views

Doubt for substitution in two variables limit

I have an exercise on my book like that $\lim_{(x,y)\to (0,1)}f(x,y)=\frac {(y^2-1)(9x^2+2)\log\left(1+x^5\right)}{x^4+(y-1)^6}$ then it says without other explanations: for $(x,y)\to(0,1)$ you have $...
0
votes
1answer
21 views

Finding the gradient vector of a plane along the plane's surface

How do you find the gradient vector of a plane? I have a plane that passes through the origin with the equation P: 5x + 95y + 46z = 0 whose normal ...
3
votes
2answers
96 views

Show differentiability of $f(x) = \sqrt{x^4 + y^4}$ in $(0,0)$

I'm trying to show the differentiability of $$f:\mathbb R^2 \to \mathbb R\text;\quad f(x) = \sqrt{x^4 + y^4}$$ in (0,0). Here's my attempt: Since $\partial_xf(x,y) = \frac{4x^3}{2\sqrt{x^4+y^4}}$ we ...
0
votes
1answer
47 views

Analytic Solutions to differential (heat) equation.

I've searched around, but I couldn't find anything too helpful on the subject, so here goes. I am trying to find (if possible) an analytical solution to the differential equation of the form: $$ \pm\...
0
votes
0answers
28 views

Inner lebesgue measure definitions

I´m currently studying lebesgue measure theory. I'm using Lebesgue integration on euclidean space by Frank Jones as one of my reference books. He defines the outer and inner lebesgue measures for an ...
1
vote
1answer
29 views

What is $f(r\cos(\theta), r\sin(\theta))$ equal to $1$ in this double integral?

Use a double integral to find the area of the region. The region inside the circle $(x − 2)^2 + y^2 = 4$ and outside the circle $x^2 + y^2 = 4$. I understand how to get the limits of integrand for ...
2
votes
2answers
20 views

Evaluate the flux of the vector field $\vec F = -9\hat j- 3 \hat k$ on the surface $z=y$ bounded by the sphere $x^2+y^2+z^2=16$

Evaluate the flux of the vector field $\vec F = -9\hat j- 3 \hat k$ on the surface $z=y$ bounded by the sphere $x^2+y^2+z^2=16$ My attempt: $$\iint_S \vec F \cdot \vec n dS = \iint_S (0,-9,-3) \...
1
vote
0answers
22 views

cycloid of a unit-speed circle

In one of the lectures of the MIT OCW Multivariable Calculus course, the professor introduces the parametric equation of a cycloid in the plane, where $a$ is the radius of the circle that creates it, ...
0
votes
0answers
23 views

Position vector, vectorfunction or vector field?

This notation confused me. The position vector in the wikipedia-article is denoted (cartesian coordinates): $$\mathbf{r}(t)=\mathbf{r}(x,y,z)=x(t)\mathbf{\hat{e}}_x+y(t)\mathbf{\hat{e}}_y+z(t)\mathbf{...
0
votes
2answers
36 views

Conceptual Question: What is the purpose of dotting with the normal in Divergence Theorem?

Consider the Divergence Theorem in this form $$\int_U \nabla\cdot F\,dV_n=\oint_{\partial U}F\cdot n\,dS_{n-1}$$ I am still not entirely sure what is the purpose of "dotting" with the normal as in $F\...
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0answers
38 views

Spivak's proof of change of variable

I'm reading the proof of change of variable in Spivak's Calculus on manifolds. In the last inequality of page 68, I think this proof assumed that $ f\circ g |\det g'| $ is integrable on $g^{-1}(V)$ . ...
0
votes
1answer
26 views

Conceptual question: If directional derivative exists, is it necessarily of this form?

Let $f:\mathbb{R}^2\to\mathbb{R}$ and let $v=(v_1,v_2)\in\mathbb{R}^2$. Question 1) Suppose the directional derivative $D_vf$ at $c=(p,q)$ exists. Must $D_vf$ necessarily be of the form: $$D_vf(c)=...
1
vote
1answer
22 views

Double integration over a general region

$\iint x^2 +2y$ bound by $y=x$ $y=x^3$ $x \geq 0$ this is either a type I or type II since the bounds are already nicely given for a type I, I integrated it as a type I: Finding the bounds: $x^3=x ...
0
votes
2answers
24 views

Where's the error in this equation of a plane

I was given the question: Find equation of a plane with $P$$(-4,-4,-2)$ and normal vector $\langle -1,4,1 \rangle$. My final answer was: $$-x+4y+z=-10$$ But the last part is wrong $(-10)$. How is ...
3
votes
1answer
51 views

Show that $σ_n$ converges uniformly to $σ$.

Let $a$ and $b$ be two points of $\mathbb{R}^2$.Let $σ_n : [0, 1] → \mathbb{R}^2$ be a sequence of continuously differentiable constant speed curves with $||σ_n'(t)|| = L_n$ for all $t ∈ [0, 1]$ and $...
1
vote
1answer
56 views

Solving coupled second order ODEs via Laplace transforms & Function theory.

I have used Laplace transforms to transform a system of 2 coupled second order ODEs into 2 simultaneous equations. 1st ode: $$\frac{3d^2y}{dt^2}+\frac{dy}{dx}=0$$ 2nd ode: $$\frac{5d^2y}{dx^2}-\...
2
votes
1answer
99 views

Proving a function is not differentiable, when its partials are not continuous

Let $$f(x,y)=\frac{y \sin (3 x)}{\sqrt{x^2+y^2}},$$ and $f(0,0)=0$. I'm trying to prove that it's not differentiable in $(0,0)$. Some my plan was to compute the limit of the definition of ...
1
vote
0answers
28 views

derivative delta function

How can I simplify $\nabla \cdot \left(\vec{f}(x)\delta_S(x)\right)$ where $\nabla \cdot$ is 3D divergence operator and $\vec{f}$ is a 3D vector valued function. The delta function $\delta_S(x)$ is ...
0
votes
0answers
30 views

Average distance between two points in a bounded region [duplicate]

How to construct the integral in calculating the average distance between two random points inside a square? Is this the same as asking the average length of all the possible line segments which can ...
2
votes
2answers
61 views

inverse function theorem and matrix square root

Define $\mathbf{f}(A) = A^2$, for $A \in \mathbb{R}^{n \times n}$. (a) Applying the Inverse Function Theorem, show that every matrix $B$ in a neighbourhood of $I$ has (atleast) 2 square roots $A$, ...
0
votes
1answer
35 views

Divergence of a vector field in an orthogonal curvilinear coordinate system

How would one go about proving the following result in $\mathbb R^3$ for the divergence of vector field $\vec F = F_i \hat e^i$ $$ \nabla \cdot {\mathbf F} = \frac{1}{h_1 h_2 h_3} \left[\frac \...
4
votes
3answers
66 views

Textbook for Multivariable and/or Vector Calculus

I'm looking for a tetxbook that covers Multivariable Calculus and/or Vector Calculus theoretically. I have done Analysis (single-variable) at the level of Introduction to Real Analysis by Bartle and ...
0
votes
1answer
29 views

find the vector equation for the intersection of a plane and sphere

$x^2+y^2+z^2=4$ and $x+y+z=3$ First I tried to parameratize: $t=x^2 \to x=\sqrt{t}$ $t=y^2 \to y=\sqrt{t}$ Then substituting those parameters into the plane to get: $z=3-\sqrt{2t}$ These three ...
0
votes
0answers
25 views

Softmax Derivation Help

I've been reading a paper that derives logistic regression from a few assumptions . Here is the link. If you go to page 5 and look at equation 18 the author claims that this essentially says the ...