0
votes
1answer
18 views

Line Integral Around a Triangle

Let $R$ be the interior of the triangle with vertices $(0,0), (4,2),$ and $(0,2)$. Let $C$ be the boundary of $R$, oriented counterclockwise. Now evaluate the integral below. $$\int_C(y+e^\sqrt{x}) ...
0
votes
3answers
51 views

Maxwell's Equations Divergence Question

$$ \left\{ \begin{align} \text{div } \textbf{E} & =0, \\ \text{div } \textbf{H} & =0, \\ \text{curl } \textbf{E} & = \frac{-1}{c} \frac{\partial \textbf{H}}{\partial t}, \\ \text{curl } ...
0
votes
1answer
37 views

Maxwell's Equations Curl Question

$$\left\{ \begin{align} \text{div } \textbf{E} &=0, \\ \text{div } \textbf{H} & =0, \\ \text{curl } \textbf{E} & = \dfrac{-1}{c} \dfrac{\partial\textbf{H}}{\partial t}, \\\text{curl } ...
4
votes
0answers
85 views

Line integrals and path independence

Consider $\textbf{F}(x,y)=\frac{-y}{x^2+y^2}\textbf{i}+\frac{x}{x^2+y^2}\textbf{j}$. Let $C_1$ be the upper half of the unit circle oriented counterclockwise, and let $C_2$ be the lower half of the ...
3
votes
0answers
35 views

Line integral parametrization

We are given the field $\textbf{F}(x,y)=(x-y)\textbf{i}+xy\textbf{j}$ and C being $\frac{3}{4}$ of a circle of radius $2$ centered at the origin traversed from $(2,0)$ to $(0,-2)$. $$\textbf{F}(x,y) ...
5
votes
5answers
213 views

Global Max and Min Problem

I'm working on a problem which asks me to find local and global extrema of the following function. $$f(x,y) = x^2y^2e^{(-x^2 - 2y^2)}$$ I went through and found all of the relevant partial ...
0
votes
3answers
80 views

Determing if $f(x,y)$ is continuous at $(0,0)$

I would really appreciate if someone could help me figure out where to start on this problem. The question is to determine if $f$ is continuous at the origin. $$\begin{equation} ...
0
votes
0answers
29 views

Surface Integrals, orientation and parametrizations.

I'm trying to solve the following problem: Integrate $f(x,y,z)=(x,y,z)$ over the surface $z=12$ $x^2 + y^2 \leq 25$ I parametrized the surface with $\sigma (r, \theta) = r \sin(\theta), r ...
4
votes
1answer
74 views

a complicated question about double improper integral

how to evaluate $$\iint_{y\ge x^2+1}{dx\,dy\over{x^4+y^2}}$$ My solution: the initial intergral $$ =2\int_0^\infty \left(\int_{x^2+1}^\infty {dy\over {x^4+y^2}}\right)\,dx = \int_0^\infty ...
0
votes
1answer
48 views

evaluating surface integral with divergence theory

If I have to calculate the surface integral of $\iint_S A \cdot n\ \mathrm {ds}$ where $A= 3zi-2xj+5x^2zk$ and $S$ is the surface of the cylinder $x^2+y^2=4$ and lying between $z=0$ and $z=4$ in the ...
2
votes
2answers
47 views

Surface Integral of a Right Circular Cone

Use a surface integral to show that the surface area of a right circular cone of radius $R$ and height $h$ is $\pi R \sqrt{h^2+R^2}$. Hint -- Use the parametrization $x=r\cos\theta$, $y=r\sin\theta$, ...
3
votes
1answer
30 views

Divergence Theorem Question

$$\iint\limits_\sum f \ d \sigma = \iiint\limits_S \operatorname{div} \textbf{f} \ dV$$ $$\operatorname{div} \textbf{f}=1+2+3=6$$ After this, we could multiply $6$ by the volume of the sphere ...
0
votes
1answer
20 views

Showing $\int_{-1}^1\int_{-1}^1(u_x^2+2u_y^2+u^2-x^2y^2u)\, dx\, dy\geq c$.

Prove that for some $c\in\mathbb{R}$: $$G(u) =\int_{-1}^1\int_{-1}^1(u_x^2+2u_y^2+u^2-x^2y^2u)\, dx\, dy\geq c$$ for every $u \in H_0^1$. I know that $$G(u) ...
0
votes
0answers
34 views

Limits in multivariate

Find four types of functions in two variables, where converting to polar coordinates seemingly tells you that the limit exists but the limit doesn’t actually exist. (By types, I mean they cannot be ...
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
41 views

Prove that These Families of Level Curves are Orthogonal

From p. 79 in Brown's and Churchill's "Complex Variable and Application": Let the function $f(z) = u(x, y)+iv(x, y)$ be analytic in a domain $D$, and consider the family of level curves $u(x, y) = ...
1
vote
0answers
44 views

Integration in polar coordinates?

Given $$ A=\begin{pmatrix} a & b \\b & c \end{pmatrix}, x=(x_1,x_2), (Ax,x)>0 $$ and $$(x,y)=x_1\cdot y_1+x_2\cdot y_2$$ I'm trying to prove that $$ \int_{-\infty}^\infty ...
0
votes
0answers
23 views

Lagrange multipliers (distance)

Find the closest point of the surface $z=xy-1$ to the origin. How would you do that with Lagrange multipliers?
0
votes
1answer
40 views

Maximization of Function with two restrictions.

Maximize $$f(x,y,z)=xy+z^2,$$ while $2x-y=0$ and $x+z=0$. Lagrange doesnt seem to work.
0
votes
2answers
45 views

Find partial derivatives of $f(x, y)=\sqrt[3]{xy}$

Let $f(x, y)=\sqrt[3]{xy}$. Find $f_x(0,0)$ and $f_y(0,0)$. Is $f$ differential at (0,0)? How can I do?
0
votes
2answers
14 views

Determining when the gradient of a function is parallel to a vector.

Let $G(x,y,z) = \left( \sqrt{x^{2} + y^{2}} - R \right)^{2} + z^{2}$. If my calculations are correct, then $$\nabla{G} = \left(x \left(2 - \frac{2R}{\sqrt{x^{2} + y^{2}}} \right), y \left(2 - ...
0
votes
0answers
35 views

Surface integral of a cube

can anyone help me out with this question on surface integrals? Evaluate $\displaystyle\iint\limits_{S}A \cdot dS$ where $A= x^2\hat{i}+ y^2\hat{j}+ z^2\hat{k}$ and $S$ is the surface of a cube. $0 ...
2
votes
2answers
29 views

About the order of integration for double integrals

I have to compute $\int\int (2x-y) \,dx \, dy $ on the domain $\{ (x,y) \in R^2 : 1\leq x\leq 4, 0\leq y\leq \sqrt{x} \}$ So mi first try is to do: $\int_0^{\sqrt{x}}\int_{1}^{4} (2x-y)\, dx \, dy ...
0
votes
1answer
15 views

Center of Mass double Integral using polar Coord.

Find center of mass given Lamina pictured: https://s3.amazonaws.com/wamapdata/qimages/qtrring.gif with inner radius of 3 and an outer radius of 7, and a density function $$\rho(x,y) = ...
1
vote
1answer
24 views

Solve system to find critical points.

Hi I have to find the stationary points for $$f(x)= x^4+y^4-(x-y)^2.$$ So far i founded the partial derivatives for $x$ and $y$. Next step is to solve this system to get my critical points: $$ ...
1
vote
2answers
17 views

Solve this PDE using a change of variable.

Solve the following PDE using a change of variable: $$ \alpha^2 \dfrac{\partial^2 z}{\partial x^2} - \beta^2 \dfrac{\partial^2 z}{\partial y^2} = 0$$ This is my attemp: Let the following change of ...
0
votes
1answer
30 views

Derivative of a path.

Define path: $s(t)= \langle 0,\cos(t),\sin(t)\rangle$ We are given that it is on the surface of $F(x,y,z)= x^2 + y^2 + z^2 -1$ Am told to find $s'(t)$ and a given point $t = (\pi/2)$. Is $s'(t) = ...
0
votes
1answer
16 views

Where do these Paths intersect

We have two paths: $r(t)=\langle cos(t),0,sin(t)\rangle $ $s(t)= \langle 0,cos(t),sin(t)\rangle$ where $t$ in $[0,pi]$ We are given that they are on the surface of $F(x,y,z)= x^2 + y^2 + z^2 -1$ ...
1
vote
1answer
28 views

Finding the marginal distribution from a joint PDF - I can't find my mistake

The continuous random vars X and Y have the joint PDF: f(x,y) = a($y^2 - x^2)$ $$ 0 \le x \le y \le \theta $$ I have found the value of a by computing the definite integral, setting to 1 and ...
3
votes
0answers
34 views

Prove Differentiation Multivariable

Given $f(x,y) = \frac{ xy^2}{x^2 +y^2}$ From defintion we know it is differentiable if: $\lim_{h\to 0}\frac{F(X+h)-F(X)-c*h}{|h|}$ exists, where $c$ is the gradient of the function. I have ...
1
vote
0answers
65 views

Intersection of ellipse and hyperbola at a right angle

Need to show that two functions intersect at a right angle. Show that the ellipse $$ \frac{x^2}{a^2} +\frac{y^2}{b^2} = 1 $$ and the hyperbola $$ \frac{x^2}{α^2} −\frac{y^2}{β^2} = 1 $$ will ...
0
votes
2answers
40 views

How to evaluate this triple integral?

How would I go about evaluating this integral? I want to change the order of integration but don't know how. $$\int_0^1\int_1^{\Large e^z}\int_0^{\log y}x\ dx\,dy\,dz$$ I'm having difficulty ...
0
votes
0answers
33 views

What does this multivariable question mean?

This question is from a 1st year 1st semester math course so I have very limited knowledge of multivariable calculus. Find the indicated derivatives, assuming that the function $f(x,y)$ has ...
1
vote
1answer
60 views

Stoke's Theorem Example - Help?

From Stoke's Theorem: \begin{equation*} \oint_c \textbf{F}\centerdot d\textbf{r}= \int\int_S (\nabla \times\textbf{F})\centerdot d \textbf{S}\end{equation*} Evaluate $\oint _C \textbf{F}\centerdot ...
0
votes
1answer
31 views

Verifying Stokes' Theorem for a given field

a friend of mine asked me for this exercise, but I'm not sure if I'm right. For the next vectorial field $A = (2y,-3x,-z^2)$ verify the Stokes' Theorem for the upper hemisphere $x^2 + y^2 + z^2 = ...
1
vote
3answers
69 views

a question about the evaluation of triple integral, I am stuck!

How to use the method of orthogonal transformation to figure out the triple integral ?. I am stuck about it! The triple integral is: $$ \iiint\cos\left(ax + by + cz\right)\,{\rm d}x\,{\rm d}y\,{\rm ...
1
vote
2answers
71 views

Calculating $\iiint_K \sqrt{x^2+y^2+z^2}\,dx\,dy\,dz$.

I need to calculate the following in cylindrical coordinates: $$\iiint_K \sqrt{x^2+y^2+z^2}\,dx\,dy\,dz$$ $K$ is bounded by the plane $z=3$ and by the cone $x^2+y^2=z^2$. I know that: ...
0
votes
1answer
21 views

Problems regarding multivariable calculus

Let $f:\Omega\to \mathbb R$ be differentiable at $x_0\in \Omega$ ($\Omega$ is a nonempty open subset of $\mathbb R^n$), let $f(x_0)=0$ and let $g:\Omega\to \mathbb R$ be continuous at $x_0$. We want ...
2
votes
1answer
46 views

Smoothness of $f(x)/(1+|f(x)|)$ where $f \in C^1(E)$ for $E$ an open subset of $\mathbb{R}^n$

(a) Show that if $E$ is an open subset of $\mathbb{R}$ and $f \in C^1(E)$ then the function $$F(x) = \frac{f(x)}{1+|f(x)|}$$ satisfies $F \in C^1(E)$. (b) Extend the results of part (a) to $f \in ...
1
vote
1answer
16 views

Solution check: integral calculation using transformations

Could somebody check my solution to the following problem? Calculate $\iint_Ey\cos(y^2-4-x)dxdy$ if $E \in ]-\infty,0]\times [0,+\infty[$ enclosed by $x=-2, y=0, y^2-4=x$. Using $u=y^2-3,\quad ...
0
votes
1answer
32 views

Continuity proof of two-variable function.

The Assignment Determine if the following function is continuous in $(0,0)$. $$f: \mathbb{R}^2 \rightarrow \mathbb{R},\begin{pmatrix}x\\y\\\end{pmatrix} \rightarrow \begin{cases} ...
0
votes
1answer
38 views

Determine if the following function is continuous in $(0,0)$.

Assignment: Determine if the following function is continuous in $(0,0)$. $$f: \mathbb{R}^2 \rightarrow \mathbb{R},\begin{pmatrix}x\\y\\\end{pmatrix} \rightarrow \begin{cases} 1& ,x≤ 0, y ...
2
votes
1answer
28 views

Newton's binomial for matrices that don't commute?

I'll give a bit of background info as to why I'm asking. I need to find the directional derivative of $f(A)=A^m$ where $m>0$ and $A$ is an $n$ by $n$ matrix with real entries. I want to do this ...
2
votes
1answer
37 views

How to find the differential of this function

we are given the function $f: \mathbb R^n \setminus \{0\} \to \mathbb R^n$ defined by: $f(x) = \frac{x}{|x|}$ Find $Df(a)$. What I did: I tried working this out from the definition. the ...
5
votes
2answers
68 views

a question about how to prove mutivariable integral, I am struggling about it!

If $f(x)$ is Riemann integrable in $[a,b]$, and then how to prove $$\int_{a}^{b} f(x_1) \, dx_1 \int_{a}^{x_1}f(x_2) \, dx_2 \cdots \int_{a}^{x_{n-1}}f(x_n) \, dx_n={1\over n!} \left[\int_a^b f(x) \, ...
0
votes
1answer
21 views

Problem with a normal

Text of problem: "Define equation of curve, in all it points normal have the following feature: length of abscissa on the $x$-axis between beginning of coordinates and intersection of normal with ...
2
votes
0answers
38 views

Verifying the Divergence Theorem for Half of a Sphere

Here is an exercise that I was assigned for homework: .......................................................... To the bottom left, I have scanned an example problem for verifying the divergence ...
15
votes
3answers
302 views

Continuity of a function in two variables

Function $f(x,y)$ is continuous in each variable separately. Prove that there exists a point where it is continuous in two variables. I do not quite understand how to act here. I know the ...
1
vote
1answer
56 views

Finding the shortest distance between two planes using Lagrange multipliers

A problem (among a list of Lagrange multipliers problems in Earl Swokowski's Calculus) states as follows: find the shortest distance between $2x+3y-z = 2$ and $2x+3y-z=4$. I can see that the ...
2
votes
1answer
92 views

Curl Proof Question

Prove the given formula. So far I have $f\textbf{F}=(f\textbf{F}_1, f\textbf{F}_2, f\textbf{F}_3)$, but I'm not sure where to go from there. Could anyone give me some pointers? Thank you.