0
votes
0answers
21 views

Hyperbolic distance

Find the hyperbolic distance between $(0; 0; 0)$ and $(0; 0; \frac12)$ in the Poincare model. Recall that the Poincare model deems $d(P_1; P_2)=\int\frac{2}{1-r^2}ds$. What about the distance between ...
1
vote
0answers
13 views

Equation of tangent plane to a surface at certain point

So I am solving for an equation of a tangent plane to surface $z=x + \ln(2x+y)$ at the point $(-1,3,-1)$. I know I need to take partial derivative of that equation and plug in the point of the ...
2
votes
1answer
31 views

Gradient Vector Question?

The temperature in some three-dimensional body is modeled by the equation $$f(x,y,z)=49-x^2-y^2-z^2$$ Find the largest rate at which the temperature is increasing when T=0. I believe this is a ...
1
vote
1answer
18 views

Tangent Planes and Surfaces (Calc 3)

I am wondering if I am on the right track for the following question: Find a for the plane $x+y+z=-1$ so that it is a tangent plane to the surface $z=x^2+ay^2$ I figured since you are given a ...
2
votes
1answer
25 views

Surface Area Line integral problem

I'm trying to figure out how to solve a surface area with surface and line integrals (showing both methods). The area I'm trying to compute is the area of the shape $$x^2+y^2=9$$ bounded by $z=0$ and ...
2
votes
3answers
63 views

Lagrange multipliers from hell

I was asked to solve this question, decided to try and solve it with lagrange multipliers as I see no other way: "Find the closest and furthest points on the circle made from the intersection of the ...
0
votes
2answers
58 views

Surface Area of Two Cylinders Calculus 3

Find the surface area of two cylinders $$y^2 + z^2 = 1$$ and $$x^2 + y^2 = 1$$ I have so far set the two equations to equal $$x= \pm z$$ and $$y= \sqrt{(1-z^2)}$$ I am a little confused on how to set ...
2
votes
0answers
39 views

Is there a generalization of integration by parts?

here is what i concerned: there are $u(x)$ and $v(x)$ in the original integration by part formula, what if the integral involve with one more function $w(x)$. Second of all, i know that there are ...
0
votes
0answers
32 views

Chain rule for several variables

I am studying this example: I follow the first two statements, but I cannot make the connection between the dot product and the derivative. Can somebody please explain how the third equation ...
0
votes
0answers
13 views

predictor-corrector method and stability

A predictor-corrector method for the approximate solution of $y'=f(t,y)$ uses \begin{equation} y_{n+1}-y_{n}=hf_{n} \tag P \end{equation} as predictor and \begin{equation} ...
0
votes
0answers
8 views

Finite difference scheme and its stability

The Finite difference scheme: \begin{equation} y_{n+3}-y_{n+1}= \frac {h}{3}(f_{n}-2f_{n+1}+7f_{n+2}) \end{equation} Deduce that the scheme is convergent and find its interval of absolute stability(if ...
0
votes
0answers
6 views

Modified Symplectic Euler

Simple harmonic motion: $y'= -z $, $z'= f(y)$ and the modified Symplectic Euler equation are $$y'=-z+\frac {1}{2} hf(y)$$ $$y'=f(y)+\frac {1}{2} hf_y z$$ deduce that the coresponding approximate ...
0
votes
2answers
27 views

Line integral over a curve in the II quadrant

I am lost here: $C = x^2 + y^2 = 4$ from $(0,2)$ to $(-2, 0)$. Calculate $ \ \int_c y^2 ds \ \ $ and give reasons the sign is correct. It's obviously the circular arc going counterclockwise from ...
0
votes
1answer
28 views

Implicit Euler Scheme and stability

Find the fixed points of the implicit Euler scheme \begin{equation} y_{n+1}-y_{n}= hf(t_{n+1},y_{n+1}) \end{equation} when applied to the differential equation $y'=y(1-y)$ and investigate their ...
0
votes
0answers
20 views

A predictor-corrector method

A predictor-corrector method for the approximate solution of $y'=f(t,y)$ uses \begin{equation} y_{n+1}-y_{n}=hf_{n} \tag P \end{equation} as predictor and \begin{equation} ...
6
votes
5answers
270 views

Why aren't these partial derivatives interchangeable?

I've ran across something that confuses me regarding multivariable functions and partial derivatives. I'll use an example to illustrate: We let $$x = f(y,t) = yt^2,$$ and define the operators ...
0
votes
1answer
29 views

Can we write $f\in C^{1}(\mathbb R^{2})$ as $f(z_{1})-f(z_{2})= (z_{1}-z_{2})\cdot G(z_{1}, z_{2})$?

Mean-value theorem for one variable, tells us that if $f:\mathbb R \to \mathbb R$ is continuously differentiable, then we can write, $f(x)-f(y) = (x-y) G(x,y)$; where $x,y \in \mathbb R$ and actually ...
0
votes
2answers
41 views

Higher Order Partial Derivatives

If i have 3 times differential function $ z= f(x^3 / y^4) $ how can i get: a) ${\partial z \over \partial x}$ b) ${ \partial ^2z \over \partial x^2}$ c) ${\partial^2z \over \partial x \partial ...
0
votes
2answers
38 views

Chain rule error

Find $\frac{ \partial ^2 f}{ \partial x ^2}$ where $f(x,y,z)=h(r)$ in $R^3$ except $(0,0,0)$ and $r$ is the usual radius. Attempt: see here $\dfrac{\partial f}{\partial x} = ...
0
votes
0answers
26 views

Using IFT to determine whether $f:(x,y)\longmapsto\left(\frac{x^2-y^2}{x^2+y^2},\frac{xy}{x^2+y^2}\right)$ has inverse function near $(0,1)$

Well, we can say $f(x,y)=(u,v)$. We want to determine whether there is a function that describes $x,y$ in terms of $u,v$. Define ...
1
vote
1answer
16 views

Iterating the chain rule in multiple variables

$$f:\mathbb{R^3}\rightarrow\mathbb{R},\quad g:\mathbb{R^2}\rightarrow\mathbb{R},\quad h:\mathbb{R}\rightarrow\mathbb{R}$$ $f,g,h$ are differentiable along their domain. I'm asked to find the total ...
0
votes
1answer
20 views

Switching $2\int_0^{2\pi} |\cos(\frac{\theta}{2})|d\theta$ to $4\int_0^{\pi} \cos(\frac{\theta}{2})d\theta$?

Context: Obtaining the arclength of path $r=\cos(\theta)+1$ (polar) using a path integral, $\theta \in [0,2\pi]$. I'm currently following a solution guide to a problem, and at one step, the author ...
0
votes
1answer
45 views

Length of a curve in $\mathbb R^n$ smaller than the distance between two points

Let $\gamma : [0,1] \rightarrow \mathbb R^n$ be s.t. $\gamma(0)=a, \gamma(1)=b$ and $\|\gamma' \|\in L^1$. How can I show that $$ \mathscr L (\gamma) = \int _0^1 \| \gamma'(t) \| dt \geq \|a-b\| ...
0
votes
1answer
22 views

Parametrization of Volume of intersection of two balls

I am trying to find a parametrization of the volume of intersection of two balls $(x-\alpha)^2+y^2+z^2 \leq R^2$ and $(x+\alpha)^2+y^2+z^2 \leq R^2$ , where $R \geq \alpha$, Also , How to find volume ...
0
votes
3answers
52 views

$x^2 + y^2 - y = 0$ is… a cylinder?

I've this question: Find the area of the intersection between the sphere $x^2 + y^2 + z^2 = 1$ and the cylinder $x^2 + y^2 - y = 0$. Is this second equation even a closed shape? If one were to ...
0
votes
2answers
28 views

Use Lagrange Multipliers to find the absolute extrema

Use Lagrange Multipliers to find the absolute extrema (if any) of: $f(x,y) = 4x^2 + 9y^2$; subject to $2x +3y = 6$. Using Lagrange I end up with one point: $(\frac{3}{2}, 1)$ I'm just not sure how ...
1
vote
2answers
47 views

Solving problems using the *definition* of differentiability

There is a problem in my textbook, that I could not solve and was not able to understand the solution to. The problem had part a, b, c, d. Only a were solved. I am out of luck. I hope, if somebody ...
1
vote
2answers
39 views

Is it possible to find a function if we know its differential?

Not something we were taught at uni yet, just something that peaked my curiosity. If I was given a derivative of a scalar function, for example $f'(x)=x$ then I know that $f(x)=\frac{x^2}{2}$ (let's ...
0
votes
1answer
17 views

double integration via u-subtitution

I'm having trouble with this double integral, maybe someone can help me out: $\int_1^2 \int_0^{lnx} 4x \ dy dx$ My attempt: $$\int_0^{lnx} 4x \ dy = 4xy \big |_{y= 0}^{y= lnx} = 4x \ln(x) $$ $$ ...
1
vote
0answers
64 views

Understanding double Riemann sums

I have the following two parts of a question, and I merely want to understand what is being asked of me: 1) Divide the region of integration into 4 rectangles of equal width and height. By ...
1
vote
1answer
57 views

Integration w/ Change of Variables

folks. I've got this question: Let $D$ be the region $\{(x,y) ~|~ 0 \leq y \leq x, 0 \leq x \leq 1\}$. Evaluate: $$\iint_D (x + y) dxdy$$ by making the change of variables $x = u + v$, $y = u ...
0
votes
2answers
27 views

higher partial derivative

I'm confused here: $$f(x,y) = \sqrt{x^2 + y + 4}$$ I got: $$\frac{\partial f}{\partial x} = x(x^2 + y + 4)^{-\frac{1}{2}}$$ $$\frac{\partial f}{\partial y} = \frac{1}{2}(x^2 + y + ...
1
vote
1answer
41 views

Find maximum on elipsoid using implicit function theorem…again

I feel like im drowning this site with question about implicit function theorem but I really do not understand how I can find the differential. we are given elipsoid $x^2+y^2+z^2+xy+yz-54=0$ We are ...
0
votes
1answer
14 views

Evaluate an integral over $\mathbb{R}^{3}$ and Green's Theorem

Let $p(x_{1}, x_{2}, x_{3})$ be a smooth function in $\mathbb{R}^{3}$ decaying sufficiently rapid as $|x| \rightarrow \infty$. Why is $$\int_{\mathbb{R}^{3}}p_{x_{i}}\, dx = 0?$$ By the Gauss-Green ...
0
votes
1answer
21 views

Find the following partial derivatives?

$$F(x,y,z)=x^8y^2+\sin(y^3z^2)+3=0$$ Find $∂z\over∂x$ and $∂z\over∂y$. I'm pretty confused since I'm only used to finding partial derivatives of something like $∂F\over∂x$ or $∂F\over∂y$. Any help ...
5
votes
1answer
78 views

Multivariable calculus - find derivative using implicit differentiation

Short simple question which i managed to solve partially. we are given the equation $x^2+y^2-z^2+xz-yz-1=0$. Show using the implicit function theorem that this equation sets in the neighborhood of ...
0
votes
1answer
32 views

Transform $f''_{xy}$ into $u$ and $v$.

Transform $f''_{xy}$ into $u$ and $v$. See paper below. I miss out on one term, I suspect that I forgot the product rule somwhere, but I cannot tell where.
0
votes
0answers
32 views

Treating partials as regular fractions

$\frac{ \partial ^2 f}{ \partial x ^2} = \frac{\partial f}{ \partial x} \cdot \frac{\partial f}{ \partial x} $ I saw my teacher just multiply two partials, to get the second derivative, se image. ...
2
votes
1answer
47 views

Lagrange multipliers with constraints

I am trying to get the max and min of the function $f(x,y,z)$ and $h(x,y,z)$ is the constraint. I have done the below steps. Are the value of my points correct for $x,y,z$? If so how do I proceed to ...
0
votes
0answers
31 views

Creating a $C^2$ smooth convex curve which joins circular arcs

Suppose that I have many points $p_1$,..., $p_n$, all very close to origin 0. For R very, very large, I have circular arcs of radius R, $\alpha_i$, about $p_i$. I do not speak about their ...
0
votes
0answers
18 views

2 competitive or cooperative products?

I found this question in a calculus book and I'm not sure how to go about solving it: What exactly am I supposed to do to answer this? Do I need to find the first and second derivatives of each ...
1
vote
1answer
12 views

Green's Theorem Conditions

Why must the following condition be satisfied before we can use Green's Theorem: L and M are functions of (x, y) defined on an open region containing D and have continuous partial derivatives there.
0
votes
0answers
10 views

Proof of a multi-variable limit

$$\text{Let }f : D \subset R \to R^2 \text{ defined by } f(x,y) = 1 $$ where $$ D = B_1(0,0) \cup \text{ {(1,0)}} \text{ and } B_1 $$ is the open disc of unitary radius centered at the origin $$ ...
0
votes
3answers
72 views

A proof for $ \lim_{(x,y) \to (0,0)} \frac{\sin(x^2 + y^2)}{x^2 + y^2} $

To prove the following limit $ \lim_{(x,y) \to (0,0)} \dfrac{\sin(x^2 + y^2)}{x^2 + y^2} = 1$ is it sufficient to say that around $(x,y) \to (0,0) $ we have $\sin(x^2 + y^2) \approx x^2 + y^2$?
0
votes
2answers
43 views

evaluate the limit or show that it doesn't exist

Evaluate the limit or prove that it doesn't exist . $$\lim_{(x,y) \to (0,0)}\frac{\sin(2x) -2x + y}{x^3 + y} $$
1
vote
1answer
25 views

Triple integral question

In a textbook problem, I am asked to find the flux through a given surface using the divergence theorem. That is, $$\iiint_{G} \nabla \cdot \vec{F} \, dV = \iint_{\sigma} \vec{F} \cdot \vec{n} \, dS$$ ...
2
votes
2answers
46 views

Finding max/min of multivariable function

The following function $f(x,y) = 3xy + \frac{6}{1 + x^2 + y^2 }$ within $\frac{1}{3} \leq x^2 +y^2 \leq 4$ I do partial differention $\frac{\partial z}{\partial x} = 3y - \frac{12x}{1 + x^2 + ...
3
votes
2answers
207 views

Integrating sin(x) on a unit circle

I am trying to integrate $\int\int_{D} sin(x)$ where $D$ is a unit circle centered at $(0,0)$. My approach is to turn the area into the polar coordinate so I have $D$ as $0\leq r\leq1$ , $0 \leq ...
0
votes
1answer
12 views

how to find family of curve with a maxima and minima

How to find the family of curve in 2D such that it has minima at two points suppose x=a and x=c, and maxima at x=b where a< b < c. I can find a single curve because y’ =0 at maxima and minima. ...
0
votes
2answers
67 views

Evaluate the limit

Does this limit equals 0 or it doesn't exist ? $$\lim_{(x,y) \to (0,0)} \frac{y^2 \sin(x)\cos(y)}{x^2 + y^2} $$ What I have done already : $$ \quad 0\leq\frac{y^2}{x^2 + y^2} \leq 1 \\ ...