1
vote
0answers
19 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
18 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
66 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
22 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
29 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
24 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
41 views
0
votes
1answer
34 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
53 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
37 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
15 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
278 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
53 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
33 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
86 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
28 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
44 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
56 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
36 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
41 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
40 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
67 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
33 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
40 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
54 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
39 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
115 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 ...
0
votes
2answers
47 views

double partial differentiation

I'm having troubles with solving problems with partial differentiations... and this one is double. I don't thing we've even learned this in class... Question: If $z=f(x,y)$, where $x=r\cos(\theta), ...
0
votes
1answer
30 views

partial differentiation of multivariable function

Let $a,b$ be constants satisfying $a^2+b^2=1.$ Let $f=f(x,y)$ be a twice differentiable function, and $u=ax-by$ and $v=bx+ay$. For the function $g$ defined by $g(x,y)=f(u,v)$, prove ...
1
vote
0answers
11 views

upper bound for error in multivariable differential

The question is, let $f(x,y,z)=x\cos(yz)$. by how much will the function $f$ change as the point $P(x,y,z)$ moves from $P_0(1,0,0)$ a distance of $0.1$ unit toward the point $P_1(1,1,1)$? Also derive ...
0
votes
1answer
15 views

Lagrange multipliers and angle between vectors

Can someone please help me with solving this question? I'm new to learning this and I'm not at all sure if what i've done is correct... The question is: the plane $4x-3y+8z=5$ intersects the cone ...
0
votes
1answer
25 views

For a 2 variable function, prove that the linear approximation is less than the real value for all x and y

Consider $f : \Bbb R^{2} → \Bbb R$ defined by $f(x,y) = x^{2} + 3y^{4}$. Prove that $f(x,y) ≥ L(x,y)$ for all $(x, y)$ in $\Bbb R^{2}.$ I found the linear approximation: $$L = f(x,y) + f'x(a,b)(x-a) ...
0
votes
1answer
39 views

Calc 3 double integral

Compute the double integral of $f(x,y)=3\sin(5x)$ over the domain $D$ bounded by $x=0, x=\frac{\pi}{10}, y=0, y=\cos(5x)$. I am having trouble solving this double integral. I know that I must go ...
3
votes
1answer
39 views

$f:\mathbb{R^2}\setminus\{(0,0)\}\ \rightarrow \mathbb{R}$ of class $C^2$ for which $f_x(x,y)=\frac{y}{x^2+y^2}$ and $f_y(x,y)=\frac{-x}{x^2+y^2}$

Is there exists $f:\mathbb{R^2}\setminus\{(0,0)\}\ \rightarrow \mathbb{R}$ of class $C^2$ for which $f_x(x,y)=\frac{y}{x^2+y^2}$ and $f_y(x,y)=\frac{-x}{x^2+y^2}$ for all ...
4
votes
2answers
47 views

limits using $ \epsilon - \delta $ to prove two variable function

I'm trying to use the $ \epsilon - \delta $ argument to prove $\lim_{(x,y) \rightarrow (1.1)} \frac{2xy}{x^2+y^2} =1$. I know that I need to show that $\forall \epsilon>0, \exists \delta>0$ ...
0
votes
1answer
39 views

What am I doing wrong in this continuity check?

I want to show that the function $f$ is discontiunous. $f$ is defined as follows: $$f(x,y) := \begin{cases} (x^2+y^2)\cdot\sin(\frac{1}{\sqrt{x^2+y^2}}) & , (x, y) \neq (0,0) \\ 0 ...
1
vote
1answer
22 views

Finding a scalar field whose gradient is a given conservative vector field

I'm studying for a course in electromagnetism, and I've been given an electric field for which I need to find the associated scalar potential. I was going to originally post this in the physics ...
1
vote
1answer
30 views

Optimization with both equality and inequality constraints

I need to minimize the following quantity: $$\min x_1^{-1/n}- \left(1-x_2 \right)^{-1/n}$$ subject to: $1-x_1-x_2=\gamma$ and $0<x_1+x_2<1$ $\gamma$ being a constant. Had it been two ...
0
votes
0answers
30 views

Area of the image of the region

I have problem which I am unable to solve. If $f:\Bbb R^2 \rightarrow \Bbb R^2$ be defined by $f(x,y)=(e^{x+y},e^{x-y})$ then find area of the image of the region $ \{(x,y):0<x,y<1 \}$
2
votes
1answer
44 views

$\int_D e^\frac{x-y}{x+y}dxdy$ Where is my mistake?

I'm trying to compute $$\int_D \! \exp\left(\frac{x-y}{x+y}\right) \, \mathrm{d}x \, \mathrm{d}y,$$ where $D$ is the region $0 \leq x \leq 1$, $0 \leq y \leq 1-x$, a triangle in the first ...
3
votes
5answers
134 views

Best Math Plotting Software for Electrical Engineering

I am an electrical engineering undergrad. I would like to learn a math plotting software which would be helpful in visualizing topics in advanced calculus (my immediate need). It would also be helpful ...
1
vote
1answer
47 views

Is the function $f(x_1,x_2) := (x_1^2+x_2^2) \sin(1/\sqrt{x_1^2+x_2^2})$ differentiable at $(x_1,\,x_2)=(0,\,0)$?

Is the function $$f(x_1,x_2) := (x_1^2+x_2^2) \sin(1/\sqrt{x_1^2+x_2^2})$$ differentiable at $(x_1,\,x_2)=(0,\,0)$? How to prove it. I have already found that the partial derivatives $\partial_{x_1}f$ ...
1
vote
1answer
19 views

turn $D=\{(x,y) | x \leq y \leq 2x, a\sqrt{x} \leq y \leq b \sqrt{x}, x\geq 0, 0 <a<b\}$ into a rectangle

Im trying to transform the region $D=\{(x,y) | x \leq y \leq 2x, a\sqrt{x} \leq y \leq b \sqrt{x}, x\geq 0, 0 <a<b\}$ to a rectangle by using a variable change. I'm doing this in order to ...