2
votes
3answers
58 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 ...
1
vote
0answers
21 views

Parametrizing a “surface” that is actually a curve, then integrating to find the area?

Find a parametrization of the surface $x^2-y^2=1$, where $x>0$, $-1 \leq y \leq1$ and $0 \leq z \leq 1$. Use your answer to express the area of the surface as an integral. I'm confused because ...
0
votes
2answers
39 views

Integrating $g: ℝ^2\to ℝ$ - Order of Integration

The problem: My work: I found the two integrals to be equal to each other, which is clearly not the desired result. Any suggestions/pointers? Thanks!
0
votes
2answers
26 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 ...
2
votes
1answer
57 views

Verify Stokes's Formula for…

Verify Stokes's Formula for $\textbf{F}(x,y,z)=(3y,-xz,yz^2)$, where $S$ is the surface of the paraboloid $2z=x^2+y^2$ bounded by the plane $z=2$. So I need to compute the integral using the formula ...
0
votes
1answer
24 views

Calculate double integral (not sure what im doing wrong)

$ \int \int (4x)/(1+xy) dA R = [0,4] \times [0,1] $ I intgrated by y first and got $(4ln(1+xy) |_(1,0))$ Which got me $4ln(1+x)$ Then integrated by x and got (4(1+x)ln(1+x) - x |(4,0) And ...
2
votes
1answer
46 views

Evaluating the surface integral?

I have to evaluate the $\iint_S z ds$ where $S$ is the part of the of the plane $5x+3y+z=15$ that lies in the first octant. I have been working on this problem for lon capa but I keep getting the ...
2
votes
1answer
37 views

Evaluate the path integral and interpret it geometrically?

Let $f(x,y)=2x-y$, and consider the path $x=t^4, y=t^4,-1 \leq t \leq 1$. (a) Compute the integral of $f$ along the path. What does this mean geometrically? My attempt: Let $\textbf c(t) = ...
1
vote
1answer
22 views

Line integral segment of parabola

Suppose $$ \vec{F} = \nabla f(x,y) = 6y \sin (xy) \vec{i} + 6x \sin (xy) \vec{j}, $$ and C is the segment of the parabola $y = 5 x^2$ from the point $(2,20)$ to $(6,180)$. Then, what is $$\int_C ...
1
vote
2answers
31 views

Volume integral help

I have a volume integral to compute with the following bounded volume $V\in \mathbb{R}^3$ $$ \frac{x^2+y^2}{4}+z^2\leq 1 \;\;,\;\; \frac{1}{2} \sqrt{x^2+y^2}\leq z\;,\;\; z\geq 0$$ I hadn't a clue how ...
2
votes
1answer
25 views

Verify integral over a surface

Show $\int \int_{S} (x^2 +y^2) d\sigma = \frac{9\pi}{4}$ where $S = \left\{(x,y,z) : x>0, y>0, 3>z>0, z^2 =3(x^2+y^2)\right\}$. We have the formula $\int \int_{S} f(x,y,z) = \int \int_{D} ...
1
vote
1answer
46 views

Line integral with Stokes

Let $C$ be curve $(x-1)^2 + (y-2)^2 =4$ and $z=4$ orientated counterclockwise when viewed from high on the z-axis. Let $$\mathbf{F}(x,y,z)=(z^2 +y^2 +\sin ...
0
votes
1answer
33 views

Cauchy-Reimann equations and harmonic functions

I have the following question which I am supposed to use the Cauchy-Riemann equations to prove: Let u and v have continuous second derivatives satisfying: $\frac{\delta u}{\delta x} = \frac{\delta ...
1
vote
1answer
24 views

Wave Equation Descending from 2D to 1D

I am stuck deriving the familiar d'Alembert formula using the Method of Descent, going from 2D to 1D. After using Kirchhoff's Formula, writing the solution independently of $y$ and some manipulations, ...
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
1answer
33 views

Basic surface integral with Stokes.

Calculate surface integral $\iint_S \nabla \times \mathbf{F} \bullet \mathbf{n} \; dS$ with stokes, when $$\mathbf{F}=\left\langle\frac{5y(z-1)}{6},xz, 6e^{xy}\cos{z}\right\rangle$$ and $S$ is surface ...
0
votes
0answers
23 views

find the volume of the region using triple integrals using cylindrical coordinates

The volume of the pyramid defined by $(0,0,0)$, $(2,0,0)$, $(0,1,0)$ and $(0,0,4)$. Calculate: $\displaystyle\iiint(2+z^2)\,dV$ The limit of the radius is where I am stumped, since the $xy$ ...
0
votes
1answer
19 views

Question with divergence theorem

Calculate flux of field $$\mathbf{F}=(3x^2 y+6)\mathbf{i}+\left(\frac{x y^2 +1}{3}\right)\mathbf{j}+(3yz^2+3)\mathbf{k}$$ in a box where it's opposite angles are in $(1,1,1)$ and $(2,2,2)$ and it's ...
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 ...
1
vote
1answer
32 views

Use Lagrange Multipliers to show the distance from a point to a plane

I'm trying to use Lagrange multipliers to show that the distance from the point (2,0,-1) to the plane $3x-2y+8z-1=0$ is $\frac{3}{\sqrt{77}}$. Our professor gave us two hints: We want to minimize a ...
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
18 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 ...
0
votes
1answer
33 views

Validity of homework question?

Is the following homework problem posed correctly? If $S$ is a sphere and $F$ satisfies the hypotheses of Stokes' theorem, show that $$\iint_{S}F\cdot ds =0.$$ My intuition tells me that the problem ...
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
19 views

Elements of a Negative Semidefinite Matrix

Use the definition of a negative definite matrix to show that if A is negative semi-definite: $$A_{ii} ≤ 0 \ \forall i $$ I know the definition (in terms of quadratic form) and the equivalent rules ...
2
votes
1answer
51 views

Question with Stokes theorem

Show with Stokes that $\oint_C (y\mathbf{i}+z\mathbf{j}+x\mathbf{k})\bullet d\mathbf{r}=\sqrt{3}\pi a^2$ when $C$ is intersection of $x^2+y^2+z^2=a^2$ and $x+y+z=0$. My work: $$z=g(x,y)=-x-y$$ ...
1
vote
1answer
128 views

Line integral using Green's theorem.

Let $C$ be parametrization $\mathbf{r}=\sin(t) \mathbf{i}+\sin(2t) \mathbf{j}$, $t \in [0, 2\pi]$. Sketch picture and investigate how the surface orientates. Calculate line integral $\oint_C ...
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 . $$\lim_{(x,y) \to (0,0)}\frac{\sin(2x) -2x + y}{x^3 + y} $$
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 + ...
1
vote
1answer
33 views

Show that $\textbf F$ is a gradient field by giving a scalar function $f$ on $\Omega^+$ such that $\nabla f=\textbf F$.

Show that $\textbf F$ is a gradient field by giving a scalar function $f$ on $\Omega^+$ such that $\nabla f=\textbf F$. $\textbf F = (\frac{-y}{x^2+y^2},\frac{x}{x^2+y^2},0)$, $\Omega^+ = ...
2
votes
1answer
21 views

Integrating over a y-simple region $D$?

Let $D=\{(x,y)\space|\space1\leq x^2+y^2 \leq 2 \text{ and }y\geq0\}$. Evaluate $\int\int_D(1+xy) dA$. So I stated that $D$ is a y-simple region because for all $(x,y)\in D$, $\sqrt{1-x^2} \leq y ...
0
votes
1answer
16 views

Determine the force F acting on the particle at time $t=\frac{\pi}2$ to keep it moving along the given curve

I understand how to find part (a). It is the norm of the velocity when $t=2$, however I am clueless as to how to tackle (b).
0
votes
2answers
66 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 \\ ...
1
vote
3answers
46 views

prove the limit using definition

Prove the limit by using the definition $$\lim_{(x,y) \rightarrow (0,0)} \frac{x^{2}y^{2}}{ \sqrt{y^2 + 1}-1 } = 0 $$
2
votes
2answers
62 views

Question about Implicit function theorem

I was asked a simple question, show that $y+\sin y=x$ sets in the neighborhood of $(0,0)$ $y$ as a function of $x$, and find $\dfrac{dy}{dx}(0,0)$ Firstly, my naive solution would be: Since $lim_{y ...
0
votes
0answers
14 views

Showing that this local max is a global max

Let $$g(x_1...x_n)=x_1^{p_1}\cdot...x_n^{p_n}$$ $$u(x_1...x_n)=p_1x_1+...p_nx_n$$ Where $\sum p_i = 1$. I have to show that $f(x)=g(x)-u(x)$ is always negative or $0$ over $\Bbb R_+^n$. I've ...
3
votes
2answers
37 views

Parametrizing a 3D surface

Find a parametrization of the surface $x^3 + 3xy + z^2 = 2$, $z > 0$, and use it to find the tangent plane at $x = 1$, $y = \dfrac{1}{3}$, $z = 0$. I know how to find the tangent plane once I have ...
0
votes
1answer
28 views

Surface integral question help.

If $S$ is the surface of the sphere $x^2 + y^2 + z^2 = a^2$, compute the value of the surface integral $$\iint_S xz\,{\rm d}y\,{\rm d}z + yz\,{\rm d}z\,{\rm d}x + x^2\,{\rm d}x\,{\rm d}y$$ ...
0
votes
1answer
29 views

Properties of a function $f=f(x,y)$ with maximum in $(x_{0},y_{0})$

I wanted to ask about the following statement. Let $f=f(x,y)$, $f$ is maximum at ($x_{0},y_{0}$). Show that: 1)$\frac{\partial f}{\partial x}(x_{0},y_{0})=0$ y $\frac{\partial f}{\partial ...
1
vote
1answer
54 views

Inverse function theorem question - multivariable calculus

This is an exercise in Inverse Function Theorem http://en.wikipedia.org/wiki/Inverse_function_theorem we are given the function $f:\mathbb R^2 \to \mathbb R^2$, $f(x,y)=(e^x \cos y,e^x \sin y)$ 1) ...
2
votes
2answers
57 views

Showing that $\phi:\Bbb R^2 \to \Bbb R^2$ is injective

I need to show that $$\phi:(x,y)\to(\sin\frac{y}{2}-x, \sin\frac{x}{2}-y)$$ Is a $C^1$-diffeomorphism. So, I need to show it's injective. How can I do this? Just explicitly setting $\phi(x, ...
0
votes
1answer
47 views

Finding the volume of a 15-dimensional unit ball [using elementary multivariable calculus]?

It wouldn't surprise me if what I am asking is impossible (knowing my professor), but is there a way to find the volume of a 15-dimensional unit ball using elementary multivariable calculus? How ...
1
vote
2answers
110 views

Show that the intersection of a plane…

Show that the intersection of the plane $z = 2y$ with the elliptic cylinder $\frac{x^2}{5} + y^2 = 1$ is a circle. Find the radius and center of this circle. Hint: How can one describe a circle in ...
0
votes
1answer
25 views

Find the parametric equations of the line of intersection…

Find the parametric equations of the line of intersection of the planes x - z = 1 and x + 2y + 3z = 1. I'm assuming it's something to do with cross product? Here's what I've set up: x y z 1 ...
0
votes
2answers
42 views

center mass of the solid

Find the center mass of the solid bounded by planes $x+y+z=1,x=0,y=0$ and $z=0$, assuming a mass density of $$\rho(x,y,z) = 10 \sqrt{z}.$$ I could not set up the integral!
0
votes
2answers
28 views

How to calculate volume of a solid under a given surface with double intergrals?

How can I calculate the volume of the solid under the surface $z = 6x + 4y + 7$ and above the plane $z = 0$ over a given rectangle $R = \{ (x, y): -4 \leq x \leq 1, 1 \leq y \leq 4 \}$? I know I have ...
-2
votes
0answers
67 views

Homework: Proving stationary point over a closed convex set.

I'm stuck on this and need some help. I know the definition of a stationary point over a set. $x \in X \implies \nabla f(x^*)^T(x-x^*) >= 0$ How do i show it? I've thought of beginning by ...
2
votes
2answers
34 views

How to plot a surface in maple where the range is given by an expression, not constants?

Im trying to plot the surface $z=(1+x^2)/(1+y^2)$ , but specifically the part of the surface that is above $|x|+|y|\leq1$. Cant seem to find any information on how to produce a plot in maple, where ...