0
votes
1answer
20 views

Calculating the area of a region using a mapping

The region: $\{{(x,y) \mid x^{2} < y < 2x^{2}, 2y^{2}<x<3y^{2}, x > 0, y > 0}\}$ The mapping: $u = y/x^{2}$, $v = x/y^{2}$ I calculated the jacobian to be $\frac 34$ which means ...
3
votes
1answer
45 views

Use a double integral in polar coordinates to find the area

So the area is just an intersection of two circles Converting the two circles to polar coordinates, I get: $r(r-2\sin\theta)=0$, and $r(r-2\cos\theta)=0$ Ummm so $r =0$ and r = $2\sin\theta$ ...
3
votes
2answers
41 views

Area of spherical cap with integrals

Given a sphere $S$ of fixed diameter $D$ (or radius $R=D/2$, it will be convenient to have both, I suppose), and a point $P$ on its surface, let's create a ball $B$ of variable radius $r$ with its ...
12
votes
0answers
118 views

Evaluating $\int_{0}^{1}\cdots\int_{0}^{1}\left\{\frac{1}{x_{1}\cdots x_{n}}\right\}^{2}\:\mathrm{d}x_{1}\cdots\mathrm{d}x_{n}$

Here is my source of inspiration for this question. I suggest to evaluate the following new one. $$ I_{n}:= \int_{0}^{1} \! \cdots \! \int_{0}^{1} \left\{\frac{1}{x_{1}x_{2} \cdots ...
1
vote
1answer
16 views

derivative after composition with linear map

Let $f: \mathbb{R}^3 \to \mathbb{R}$ be a polynomial function and let $T: \mathbb{R}^3 \to \mathbb{R}^3$ be an invertible linear map. If $\nabla f(P) \neq 0$ for all $P \in \mathbb{R}^3 - \{0\}$ does ...
0
votes
1answer
35 views

(Dis)continuity of function in $R^2$

$$f(x,y) = \begin{cases} a+2x^{2}-b(y-c), & x^{2}>2+x\wedge y<6\\ 3+cx-y, & else \end{cases}$$ $f(x,y)$ is continuous on $R^2$ if $a=-3, b=1, c=2$ I think it's true: insert ...
2
votes
2answers
43 views

Frenet-Serret formula proof

Prove that $$\textbf{r}''' = [s'''-\kappa^2(s')^3]\textbf{ T } + [3\kappa s's''+\kappa'(s')^2]\textbf{ N }+\kappa \hspace{1mm}\tau (s')^3\textbf{B}.$$ What is $\tau$, I can't figure that part out. ...
1
vote
1answer
29 views

Parametric equation of a circle given starting point.

Find the parametric equations of a circle with radius of $5$ where you start at point $(5,0)$ at $v=0$ and you travel clockwise with a period of $3$. So, I know that I require to have a $x(v)$ and ...
-1
votes
1answer
19 views

Nearest and farthest point from a function to another [closed]

Find the nearest and farthest point from the ellipse $ x^2 + 3y^2 =3 $ to the segment made by $ x+y = 3 $ in the first quadrant. Found in a multivariable calculus course. So I have to find the ...
-4
votes
0answers
26 views

Triple intergrals with Volume [closed]

This question is a continuation of number 2 from the questions requiring deeper understanding of this homework. Suppose that you do not know the values of $a, b$, and $c$. Generalize what you did in ...
1
vote
1answer
21 views

Double integral calculation where $x=(y-1)^{2}-1$ and $y=x$. Not sure whether I should do it in terms of $y$ or $x$?

This is what it looks like: My first strategy was to separate it into two by drawing a vertical line at x=0 and calculate the first half in terms of x first, and the second half in terms of y ...
1
vote
1answer
72 views
0
votes
1answer
27 views

Show this integral operator is compact for various values of $\alpha$

I am having some problems evaluating a multivariable integral. This question is features in Stakgold's book Green's functions and boundary value problems. page 359. Consider the kernel for $a\leq ...
0
votes
0answers
31 views

Iterative maximization of a multivalued function - Is it possible?

I wonder if such a maximization technique possible for a multivariable, well-behaving function like $f(x_1,x_2,...,x_N)$ starting at an arbitrary point $(p_1,p_2,...,p_N)$: We first the take the ...
0
votes
1answer
32 views

Clairauts “equality of mixed partial derivatives” theorem (interpretation)

So I know how to prove this theorem via limits or whatever and I'm okay with that. What I'm not okay with is the interpretation. I just can't visualise how this is true in 3d space, any ideas? How do ...
2
votes
3answers
148 views

Trouble computing this double integral

$$\iint_R xe^{xy}~\mathrm{d}A \qquad 0\le x\le 2 \quad 0 \le y \le 1$$ Today I started learning about double integrals on a class I am taking, had good understanding on single-variable integrals but ...
2
votes
1answer
30 views

Using Green's theorem, with holes in region

I've just learned Green's theorem and I need a little help in solving a problem! I need to calculate $\oint_c \vec{F} d \vec{r} $, when the vector field $ \vec{F} =( \frac{y}{x^2+y^2}+ ...
1
vote
0answers
25 views

A calculus problem regarding mass

I am reviewing calculus and working on the following problem. In $\mathbb{R}^{3}$, the density function is given by $\mu(x, y, z) = |z|$ and if the region $R$ is given by $R : 2 \leq x^{2} + y^{2} + ...
0
votes
2answers
20 views

Line integrals; How to set $t$ boundary?

I'm having a hard time understanding how to set t boundaries in line integrals... The question is: find the line integral of $f(x,y,z)$ over the straight line segment from $(1,2,3)$ to $(0,-1,1)$. I ...
1
vote
3answers
69 views

Evaluating the integral of an exact differential

What is wrong with evaluating the closed path integral as the following? $$ \oint_\gamma \frac{x\,dy-y\,dx}{x^2+y^2}= 2\pi\ne\oint_\gamma d\left(\arctan\left(\frac{y}{x}\right)\right)=0 $$ where the ...
1
vote
1answer
24 views

Line integrals and parametrization

I've just learned about line integrals, and I need some help understanding an example problem in my textbook. The question is supposed to be really easy. Integrate $f(x,y,z)=x-3y+z$ over the line ...
2
votes
1answer
31 views

What is a closed form expression for the ∂/∂w(∂t/∂w) if w(t) is complicated function?

Lets say we have a trigonometric function w(t) that can not be inverted as t(w). The derivative ∂t/∂w can be calculated as 1/(∂w(t)/∂t). What is a closed form expression for the second derivative ...
0
votes
1answer
25 views

Finding the image of a region transformed by a mapping

The only examples I've found are either very complicated, or state the transformation like y=g(u,v) x=f(u,v), whereas this question states u and v in terms of x and y. I'm not sure how to get ...
1
vote
1answer
48 views
0
votes
1answer
36 views

Is there a formal proof for this theorem??

There is a theorem in the book Advanced Calculus by Wilfred Kaplan which states the following: The differential formula : $$ dz = \frac {\partial z}{\partial x} dx + \frac{\partial z} {\partial y}dy ...
0
votes
3answers
62 views

Find $\int_0^4\int_{0}^{4}xy \sqrt{1+x^2+y^2} \,dy\, dx $

I am having a tough time figuring this one out. Any help will be appreciated. do we have to approximate, or can we actually find it
2
votes
2answers
32 views

Double integration:$ \int_0^a \int_0^b e^{max(b^2x^2,a^2y^2)}dydx $

I would be grateful for a little help if someone could help me solve a problem in my textbook. The question is, evaluate $ \int_0^a \int_0^b e^{max(b^2x^2,a^2y^2)}dydx $, where $a,b$ are positive ...
0
votes
0answers
22 views

Average value of function over sphere

Here is another qual problem. Suppose $f:\mathbb{R}^3 \mapsto \mathbb{R}$ is $C^2$, and define the (scaled) average function $A(r)=\int_{S^2} f(rn) \:d\sigma(n)$, where $\sigma(n)$ is the usual ...
0
votes
1answer
41 views

Changing order of integration in cylindrical coordinates

I'm having a problem in changing order of integration in triple integration, in cylindrical coordinates. I would be grateful for a little help.The question is: Let D be the region bounded below by ...
0
votes
0answers
16 views

Question about the formal justification of nondimensionalization

Assume I have the following (very simple) problem: $\frac{\partial f}{\partial \theta} =0 $ and I want to make a change of variables to make it nondimensional. So, I can write: $ F = \frac{f}{f^*}$ ...
1
vote
2answers
285 views

Boundary under transformation of a closed curve from $R^2\to R^3$

Consider some mapping $\phi: R_{uv} \to S\subset \mathbb{R}^3$ where $R_{uv}\subset \mathbb{R}^2$ and such that it is a simply connected region. We call the boundary of the surface (which we ...
2
votes
0answers
61 views

Simplifying a Vector Integral

While reading the book - Cercignani, Theory and Applications of Boltzmann Transport Equation (I am not a math student), I found this integral which I am unable to understand. Note that $\xi_i , \xi_l$ ...
0
votes
1answer
35 views

Arc Length with Vector-Valued Functions, Part B

Consider the path of a particle in a conservative force field represented by the vector-valued function $$r(t)= \left(4(\sin t−t \cos t), 4( \sin t+t \sin t), \frac{3}{2} t^2 \right).$$ A) Find the ...
3
votes
2answers
89 views

Why absolute values of Jacobians in change of variables for multiple integrals but not single integrals?

If $g:[a,b]\to\mathbf R$ is a change of 1D coordinates, then the formula is: $$ \int_{g(a)}^{g(b)}\,f(x)\,dx = \int_a^b\,f(g(t))\frac{dx}{dt}\,dt. \qquad\text{(1)}$$ If $T=\{x=f(u,v); y=g(u,v)\}$ ...
1
vote
0answers
33 views

Finding maximum rate of change of total derivatives

consider $PV =nRT , P,V,T =$ pressure , volume , temperature respectively. $nR =$ constant let $n=R=1$ differentiate with respect to $t$ (time) $dP/dt = ∂P/∂T * dT/dt + ∂P/∂V * ...
1
vote
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
votes
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 ...
0
votes
2answers
38 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 ...
0
votes
2answers
42 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 ...
0
votes
3answers
48 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 ...
5
votes
5answers
72 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. ...
1
vote
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 ...
0
votes
1answer
43 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*} ...
1
vote
1answer
55 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 ...
1
vote
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
vote
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 ...
0
votes
1answer
40 views

Use divergence theorem to find $\iint_S (2x+2y+z^2) dS$ Where $S$ is the sphere $ x^2+y^2+z^2 = 1$

I tried a lot but it gets ugly really soon, any help will be greatly appreciated. T hanks
0
votes
2answers
34 views

Error in linearization: What is M in $| E(x,y) | \leq \frac{1}{2} M( |x-x_0 |+ |y-y_0 |)^2$?

I've been learning about linearization in multivariable calculus. $f(x,y) \approx L(x,y)=f(x_0,y_0)+f_x(x_0,y_0)(x-x_0)+f_y(x_0,y_0)(y-y_0)$. and the error in this is $| E(x,y) | \leq \frac{1}{2} ...
3
votes
1answer
59 views

Interesting dilemma, answer not matching with stewart, My work is Included

Question : Compute flux through the upper hemisphere of $x^2+y^2+z^2 = 1$ . Where $$\textbf{F} = \left( z^2x\right)\textbf{ i }+\left[\dfrac{1}{3}y^3+ \tan z\right]\textbf{ j } + \left(x^2z+y^2 ...
0
votes
3answers
136 views

Is speed a function of position?

Let $x$ be a smooth function from $[0,\infty)$ to $\mathbb{R}^n$ satisfying the following differential equation $x''(t) = f(x(t))$, where $f$ is a smooth function from $\mathbb{R}^n$ to itself. Then ...