3
votes
2answers
48 views

How do I evaluate this integral by hand?

TL;DR how do I evaluate $\int_0^{2 \pi } \frac{1}{\cos ^2(\theta )+1} \, d\theta$ by hand? I'm trying to solve this problem: Find the volume of the region defined by $x^2+xy+y^2+yz+z^2\le1$. ...
2
votes
2answers
49 views

Show that $f(x,y)= \|x-y\|_2^2$ is differentiable

Problem: Show that $f: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ with $f(x,y)=\|x-y\|_2^2$ is differentiable and compute its differential at every point in the domain of $f$Note: $\| \cdot ...
0
votes
1answer
33 views

The volume of a cone whose length of its side is R

How can i compute the volume of a cone whose length of its side is $R$ and the vertex of the cone forms an angle $2 \theta$ . The top cone is a cap of a sphere of radius $R$. Some help please.
-1
votes
1answer
19 views

multivariable problems

Brand Z's annual sales are affected by the sales of related products X and Y as follows: Each $\$1 $ million increase in sales of brand X causes a $\$2.1$ million decline in sales of brand Z, whereas ...
2
votes
1answer
10 views

Surface integral domain to surface function…

In a surface integral where the domain of the surface is its projection on a plane, how can there be a function from the projection to the surface if the area of the surface is greater than its ...
0
votes
1answer
22 views

Show that the image of a zero measure set is of zero measure

I saw a topic on the subject but I did not quite understand, and it was a bit old and I didn't want to resurrect it. I am going in the right direction, I just need a little nudge. let $f: \mathbb ...
0
votes
1answer
29 views

Finding/approximating 2 unknowns using one equation

I’m doing experimental data in a chemistry lab and I have faced this mathematical problem at a point of my work. Hope you guys can help me with that. What would be the best way to find two constants m ...
1
vote
0answers
38 views

Line integral - should I parametrize the square?

I have the following $1-\text{form}$ defined: $$\omega = \displaystyle\frac{2xy}{(1-x^2)^2+y^2}\mathrm{dx}+\displaystyle\frac{1-x^2}{(1-x^2)^2+y^2}\mathrm{dy}$$ I'd like to find ...
2
votes
0answers
37 views

Green's Theorem

Hey guys I am having difficulties in problem 5. I thought I understood it, but I suppose I was mistaken. I will now explain what I planned to do to solve this problem and where I got stuck. So I ...
0
votes
0answers
11 views

solving problems with joint cost and demand equations?

I have this question: I'm not sure what I need to do to solve it. I know that it will involve partial derivatives which I do not mind, I'm just not sure how to go about it. Where do I start? Do I ...
0
votes
1answer
26 views

Stokes' Theorem and Surface Independence Failure

As we know, if $\vec{F}=\nabla\times\vec{A}$ then from Stokes' Theorem, $\iint_{S_1} \vec{F}\dot \,d\vec{S}=\iint_{S_2}\vec{F}\dot \,d\vec{S}$ where $S_1$ and $S_2$ have the same boundary. Does ...
0
votes
2answers
29 views

why are conservative vector fields curl-free?

The book told me that, if a vector field $\vec{F} = Mi + Nj$ is conservative, then $$ M_y = N_x $$ But why is this true?
2
votes
1answer
44 views

How I can refine the proof of $\mathop {\lim }\limits_{(x,y)\, \to \,(3, - 1)} \left( {{x^2} + {y^2} - 4x + 2y} \right) = - \,4$

To prove that $$\mathop {\lim }\limits_{(x,y)\, \to \,(3, - 1)} \left( {{x^2} + {y^2} - 4x + 2y} \right) = - \,4$$ I followed the following process: Because the hypothesis and the definition of ...
0
votes
2answers
40 views

Absolute maximum and minimum

I am given this: $$ f(x,y) = (4x-x^2) \cos \; y \; ;\; 1 \le x \le 3 , \; \frac {-\pi}{4} \le y \le \frac {\pi}{4} $$ I did the steps below, $$f_x(x,y)=(4-2x)\cos y$$ $$f_y(x,y)=-(4x-x^2)\sin y$$ ...
0
votes
1answer
19 views

Volume of largest closed rectangular box

I came across a question from this link: http://home.educities.edu.tw/ck870522/ad05.pdf, question 28, to be specific. I read through the answer, but i don't understand their explanation.Is there any ...
1
vote
1answer
40 views

What is the limit of this function as $(x,y)$ approaches $(0,0)$?

Let the function $f \colon (\mathbf{R}^2 \setminus \{(x,y) \in \mathbf{R}^2 \colon x+y = 0 \}) \to \mathbf{R}$ be defined as follows: $$ f(x,y) \colon= \frac{xy}{x+y}$$ if $(x,y) \in \mathbf{R}^2$ ...
1
vote
0answers
33 views

To show a function differentiable

Let $A \in \mathbb{R}^n$ be a fixed vector and $T : \mathbb{R}^n \rightarrow \mathbb{R}^n$ a linear transformation . Define $f : \mathbb{R}^n \rightarrow \mathbb{R}$ by $$f(x) = \langle ...
0
votes
0answers
31 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
17 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
39 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
26 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
33 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
77 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
62 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
41 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
33 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
14 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
10 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
33 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 ...
1
vote
1answer
30 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
21 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
276 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
31 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
45 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
39 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
27 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
46 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
24 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
32 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
50 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
40 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
18 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
58 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 ...