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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What is the mean value theorem for the Fréchet (total) derivative?

What is the mean value theorem for the Fréchet (total) derivative? Off the top of my head, it's something like $$ \|F(x+h)-F(x)\|\leq \sup_{c\in[0,1]} \|F^\prime(x+ch)\|\|h\| $$ but the double ...
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1answer
19 views

Compute the volume element in a differentiable manifold.

Let $A\subset\mathbb{R}^n$ open $g:A\to\mathbb{R}$ of class $C^{1}$ and $g'(x)\not=0$ in each $x\in A$ then I want to compute $dV$ in the differentiable manifold $ M = g^{-1}(0)$. The thing is that ...
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0answers
9 views

Laplace-Beltrami on a Curve

Is there a way to write out Laplace-Beltrami operator explicitly for a sufficiently smooth plane curve given by implicit equation $s(x,y)=0$? I know that if we knew the parametrization of the curve, ...
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1answer
18 views

Help with Lagrange multipliers on an intresting function

Hi guys I am trying to do Lagrange multipliers to figure out $\lambda$ $$F=a \log(x^2-y)+b\log(x^3-z)-\lambda (x^2-y+x^3-z -1)$$ Where a and b are constants and we have the constraint $x^2-y+x^3-z ...
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1answer
26 views

I need some help understanding proofs for an upside-down cycloid being the tautochrone curve. Could someone show me or point me to a simple proof?

The tautochrone curve has fascinated me since I first heard about it and I want to share it with my Calculus class as an end of the year project. I think something similar to this (Demonstrating that ...
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0answers
30 views

Change the subject of a formula [on hold]

$150 \cdot 10^6 = \dfrac{3pR^2}{4t^2}$ How do I find out what $t$ is, hence make it the subject of the equation. I think I know what the answer should be: $p=1.5 \cdot 10^6$ $R= 0.075$ ...
3
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3answers
217 views

double integral $\int_0^t \int_0^s \frac{\min(u,v)}{uv} \, dv \, du$

I want to calculate the double integral: $$\int_0^t \int_0^s \frac{\min(u,v)}{uv} \, dv \, du$$ I don't know how to o that even if it seems simple. Thanks in advance for your help
3
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0answers
32 views

Does every irreducible projective cubic curve have a nonsingular point of inflection?

Does every irreducible projective cubic curve necessarily have a nonsingular point of inflection? I've been trying to construct counterexamples, to no avail, which leads me to believe the question can ...
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1answer
29 views

Change of variable of system of ODE [on hold]

I have one problem with the change of variables of this system: \begin{cases} 2y’ + z’ –y + 2z = 0 \\ y’ + 3z’ –3y +z = 0 \end{cases} with initial values $y(0) = 1$, $z(0) = 0$ I've made this ...
2
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1answer
57 views

integrate this double integral by any method you can. [on hold]

I'm having trouble with this double integral: $$\int_0^2\int_0^{2-x} \exp\left(\frac{x−y}{x+y}\right)\text dy\,\text dx$$
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2answers
67 views

How to find this limit $\lim\limits_{(x,y) \to (1,1)} \frac{y-x^4}{y^3-x^4}$ [on hold]

How would I find this limit? $$\lim_{(x,y) \to (1,1)} \frac{y-x^4}{y^3-x^4}$$
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1answer
20 views

Continuity of multivariable functions

I have a question regarding norms on $\Bbb R^{n}$ and proving the continuity of multivariable functions. Specifically, suppose we have $f: \Bbb R^{2} \to \Bbb R$, for example. To prove $f$ is ...
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2answers
27 views

Parametrization of an intersection cylinder ellipsoid

I'm trying to parametrize the surface given by the equations : $$\frac{x^2}{2}+\frac{y^2}{2}+z^2=1$$ and $x^2+y^2=y$. I found this function : $f:[0,1] \times [0,2\pi] \to \mathbb{R}^3$, $$(r,x) ...
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1answer
53 views

Calculate this double integral [on hold]

Recently took and exam and this was one of the questions and I wanted to check if I did it right Let $R$ be the triangular region in the ($x$,$y$)-plane with vertices $(0,0)$, $(1,0)$ and $(1,2)$. ...
1
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1answer
33 views

Partial derivative of function $\mid xy \mid + \sin{xy}$

I need to consulte one problem, just to control my result and see if I'm/ I'm not right: I want to find $$\frac{\partial f}{\partial x}(0,0), $$ where $f(x,y) = \mid xy \mid +\sin{xy}$ for $x,y \in ...
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2answers
67 views

How do ideas in differential geometry expand upon ideas from introductory calculus

I just went through first year in mathematics and used Stewart's book for calculus. I am trying to self study differential manifold and I find many concepts such as chart, atlas very similar to that ...
2
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1answer
26 views

How can I find these partial derivatives?

I'm reading a book which gives this function $f(x,y)=x^2y/(x^2+y^2)$ if $(x,y)\neq (0,0)$ and $f(0,0)=0$ as a $C^1$ function in $\mathbb R^2-\{(0,0)\}$, continuous in $(0,0)$ and it has the partial ...
0
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1answer
50 views

find the area in the first quadran

By first making an appropriate transformation, find the area in the first quadrant which is bounded by the curves $x = y, x = 2y, xy = 1$ and $xy = 2$ answer $u=x \times y$; $v=\frac{y}{x}$ (is ...
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0answers
38 views

First fundamental theorem of calculus for line integrals [on hold]

Please, could someone look at this tricky question? Find the work done by force $F(x,y)=(3y^2+2) \hat i+16x \hat j$ in moving a particle from $(-1, 0)$ to $(1,0)$ along the upper half of the ellipse ...
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1answer
19 views

Prove that Euler's equation can be written in a specific form

According to my notes, the following theorem holds: If $y$ is a local extremum for the functional $J(y)= \int_a^b L(x,y,y') dx$ with $y \in C^2([a,b]), \ y(a)=y_0, \ y(b)=y_1$ then the extremum $y$ ...
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0answers
9 views

Taylor series for a multivariable function

We know the following: $a \approx 1 + x\sqrt{dt}$ $V_1 = V(aS, t + dt ) $ The textbook claims you can (using Taylor's Theorem), expand the bottom-most equation like this: $V_1 \approx V + ...
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1answer
51 views

Sketch this body and calculate the mass of the body

A solid body in the first octant is bounded by the planes $z = 0$, $y = 0$, $z = y$ and the parabolic cylinder $y = 4 − x^2$, and has density $\rho = xz$. calculate the mass of the body. answer ...
2
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1answer
49 views

Evaluating a triple integral by inspection

I would like to evaluate the triple integral: $$\iiint\limits_D {2 + 3{x^2} + 3{y^2}dV}$$ where $D$ is a conic domain with vertex $(0,0,b)$ and axis along the $z$-axis with a base (disk) with radius ...
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0answers
28 views

Nonnegative harmonic functions

Suppose $U \in \mathbb{R}^n$ is an open domain, and $u\in C^2(U) \cap C(\bar{U})$ such that $\Delta u = 0$ in $U$. I'm working on a couple of problems pertaining to the mean value formula/harmonic ...
2
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1answer
21 views

Counterexample to the double integral computational theorem when the double integral existence assumption is dropped?

To make things simple, consider the simplest case of the double integral computational theorem. Throughout any phrase involving integrability is in the Riemannian sense. Let $[a, b], [c, d] \subset ...
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1answer
20 views

Volume inside region delimited by surfaces $z=5-x^2$, $z=y$ and $y=1$.

I need to find the volume inside the region $E$ delimited by surfaces $z=5-x^2$, $z=y$ and $y=1$. I've spent few hours on this and would really need a hint from a charitable soul. I see that the ...
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2answers
41 views

I need help with the integration order please

the integral is as follows: find the volume between these regions bounded by : $z = x^2 + 3y^2$ and $z = 9 - x^2$ I discovered that this would be the space bounded by the elliptic paraboloid and the ...
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0answers
7 views

Certain local inequality for volume and surface measures

Suppose $S$ is closed simple piecewise smooth curve for in the plane (It is viewed as boundary of a domain). Does the following hold ...
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0answers
28 views

Calculating the magnetic vector potential

I've calculated A as proportional to r^2 for ra but I really don't think is correct. If someone could take me through the calculation I would really appreciate it.
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0answers
17 views

Using polar coordinates in this integral

I'm trying to solve something along the lines of: $$\iint \frac{\partial F_1(x,y)}{\partial x}+\frac{\partial F_2(x,y)}{\partial y}dydx$$ which I want to change to polar coordinates, but I don't ...
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1answer
35 views

Critical Point of $\mathbb{R}^2 \to \mathbb{R}^2$ function

Given a function $f:\mathbb{R}^2 \to \mathbb{R}$ I can find critical points by finding the $1\times 2$ Jacobian matrix, setting each partial derivative equal to zero and solving the equations. I can ...
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1answer
18 views

Trying to integrate the volume of a body

I was trying to integrate the volume of a body blocked by $z=0$, $z=2x$, $x+y=3$ and $y=0$ using the double integral... but I don't really know how to approach this.
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0answers
41 views

Integration of a symmetric function

I have a bit of confusion about the following situation. Let's assume that we have a symmetric function $f(x,y)$ where it has the property $f(x,y) = f(y,x)$ for all $x$ and $y$. $x$ and $y$ have the ...
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2answers
38 views

Volume by double or triple integral?

I was trying to integrate the volume of a body blocked by $z=0,\; z=2x,\; x+y = 3$ and $y=0$ using the double integral... however it didn't work yet. I'm convinced its a double integral and not a ...
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0answers
14 views

Proving that average value of $u$ around a circle is the value of $u$ at the centre.

I would like to prove that: If $u(x,y)$ is harmonic in a domain containing a disk of radius $r$ with boundary $C_r$ $\implies$ the average value of $u$ around the circle is the value of $u$ at the ...
2
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1answer
36 views

Trying to integrate $\iint_D x^4\tan(x)+3y^2 \,dA$.

I'm trying to integrate $\iint_D x^4\tan(x)+3y^2\, dA$ in domain $D=\{(x, y) \in \Bbb R^2 \mid x^2+y^2\le4, y\ge0\}$. Domain is simple enough; half circle of radius 2 over $x$ axis. Converting to ...
0
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1answer
27 views

Partial derivatives after a change of variables

Say I have a function of $n$ variables $F(x_{1}, x_{2}, x_{3},...,x_{n})$, where $x_{1} = g_{1}(y_{1}, y_{2}, y_{3},...,y_{m})$, $x_{2} = g_{2}(y_{1}, y_{2}, y_{3},...,y_{m}),\dots, x_{n} = ...
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0answers
23 views

Maximizing the following function

I need to find values of $k_1$, $k_2$ and $k_3$ that maximize $C^{m_1}_{mm_1} \cdot C^{m_2}_{k_1-mm_1} \cdot C^{n_1}_{nn_1} \cdot C^{n_2}_{k_2-nn_1} \cdot C^{p_1}_{pp_1} \cdot C^{p_2}_{k_3-pp_1}$ ...
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1answer
65 views

Understanding a step in Yi Fang's Lectures on Minimal Surfaces

In Yi Fang's Lectures on Minimal Surfaces, page $94$, there's a step that I didn't understand, and that perhaps is wrong. I'll estabilish some notation first. We have that $X$ is a minimal surface, ...
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3answers
55 views

Prove that $f(x,y)$ is continuous in $(0,0)$

Prove that $f(x,y)$ is continuous in $(0,0)$, where \begin{equation} f(x,y) = \begin{cases} \frac{x^2y}{x^4+y^2}, & (x,y)\neq 0\\ 0, & (x,y) = (0,0) \end{cases} \end{equation} The solution I ...
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0answers
11 views

formula to establish correlation between multiple library functions

I am trying to predict the change in timeliness of holds delivery relative to number of owned Bestsellers, number of holds and the checkout window. (yes, it really is a library question). To do this ...
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0answers
29 views

How to prove the relation $\tan \theta = \hat{\vec{n}} \cdot \nabla h$

The relation for finding the contact angle is often given as $\tan \theta = - \hat{\vec{n}} \cdot \nabla h$ in papers such as in Sequential deposition of overlapping droplets to form a liquid line ...
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0answers
25 views

Find the mass flow rate, given a surface, density and velocity field

I have a confusion, I hope you can help me (I'd like that if you will respond, please read all my post). They ask me to find the mass flow rate passing through a surface, where the velocity field is ...
4
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1answer
119 views

Changing order of integration for the triple integral $ \int\limits_{0}^{2} \int\limits_{0}^{2z} \int\limits_{y}^{2y} f_{(x,y,z)}\; dx\, dy\, dz $

I need to change order of integration for the following triple integral: $$ \int\limits_{0}^{2} \int\limits_{0}^{2z} \int\limits_{y}^{2y} f_{(x,y,z)}\; dx\, dy\, dz $$ The domain of integration is ...
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0answers
10 views

Finite cover of balls for a bounded subset

Suppose X is a bounded subset of a k-dimensional manifold $M$ $\subset$ $R^n$ with $(k-1)$-dimensional volume $0$. I need to show that this implies for all $d>0$, all points of X can be contained ...
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1answer
52 views

Vector calculus identity for $\nabla\times(\vec{b}\cdot\nabla)\vec{b}$

I'm going through a paper on turbulence and in it the author uses the following $$ ...
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2answers
31 views

Convergence of a double integral

Is the integral $$\int_1^\infty\int_{e^{-x}}^1\frac{\sin y}{x^2y}dy dx$$ convergent or divergent?
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39 views

surface integral of the normal component of $curl(\mathbf{F})$

I have a square with side lengths $a$ in the $xy$-plane, and we build he square up vertically in the $z$-direction to construct a cube. What is the surface integral of the normal component of the ...
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1answer
24 views

Finding domain, range, and level curves of $f(x,y)=\arcsin(6y-5x)$

I will like it if someone helps me out and also checks my work for me for this equation: $f(x,y)=\arcsin(6y-5x)$ Domain $-1\leq6y-5x\leq1$ $y\geq\frac{5x}{6}-\frac{1}{6}$ and ...
2
votes
1answer
36 views

$f(x,y)=4x^3y^2$ Directional Derivative…

Let $f(x,y)=4x^3y^2$ How do I find the directional derivative of $f$ at $(2,1)$ in the direction of the vector $3i-4j$? What would be a unit vector in the direction in which $f$ decreases most ...