0
votes
1answer
16 views

continuity single variable function and multivariable funtion and its parcial derivatives

Is f(x)=1/x discontinuous at point x=0 or not since its domain is x>0 and x<0? And what about f(x,y)=$\frac{xy^2}{x^2+y^2}$ continuity? And Df(x,y) exist or parcial derivatives are ...
-7
votes
0answers
59 views

Can a set in $\mathbb{R}^2$ be closed but unbounded?

Today I read "on a closed, bounded set $D$". How can a set be closed but not bounded?
0
votes
0answers
25 views

Sufficient Conditions for Multivariate Decreasing Function

I found the following helpful theorem concerning decreasing functions but it's only valid for $\varphi:\mathbb{R}\rightarrow \mathbb{R}$, I'd like to know if it can be extended to the ...
-2
votes
1answer
32 views
2
votes
2answers
65 views

Which book is appropriate for a Chemistry student that needs to learn basics about integrals?

A friend of me who is not studying mathematics now needs to deal with integrals, double integrals and triple integrals within his study of chemistry. He asked me to give him a suggestion for a basic ...
2
votes
2answers
42 views

Find the work done by the force field in moving the particle from one point to another

Find work done by the force field F in moving the particle from $(-1, 1)$ to $(3, 2)$ This sounds good till we are given that $\textbf{F} = \dfrac{2x}{y}\textbf{ i }- \dfrac{x^2}{y^2}\textbf{ j }$ ...
-1
votes
0answers
35 views

For a map $f :\mathbb R^m\to\mathbb R^n$, prove that $\lim_{x\to a} f(x)=L$ if and only if $\lim_{x\to a}\|f(x) − L\|=0$

Could anyone help me with this proof? Given a map $f : \mathbb R^m \to\mathbb R^n,$ prove that $\lim_{x \rightarrow a} f(x) = L$ holds if and only if $$\lim_{x ...
1
vote
3answers
23 views

Gradient of modulus of vector.

I came across this in my lecture notes: This is using index notation, non-bold r is the modulus of r, and the partials are with respect to the components of r. I understand most of the steps, but ...
1
vote
1answer
57 views

Is it true that $d\textbf{S} = dy dz\textbf{ i }+ dx dz\textbf{ j }+ dx dy\textbf{ k }$

I came up with this in my mind, Just wondering if it is true I am thinking about it too, will post my observations, if any
1
vote
2answers
57 views

Evaluating the limits $\lim_{(x,y)\to(\infty,\infty)}\frac{2x-y}{x^2-xy+y^2}$ and $\lim_{(x,y)\to(\infty,8)}(1+\frac{1}{3x})^\frac{x^2}{x+y}$

I got the following problem: Evaluate the following limits or show that it does not exist: $$\lim_{(x,y)\to(\infty,\infty)}\frac{2x-y}{x^2-xy+y^2}$$ and ...
0
votes
1answer
42 views

Stokes' Theorem and Surfaces

Stokes' Theorem states the following: \begin{equation*} \oint_c \textbf{F}\centerdot d\textbf{r}= \int\int_S (\nabla \times\textbf{F})\centerdot nd \textbf{S}\end{equation*} for a given C that is the ...
0
votes
1answer
26 views

Triple integration, a general question

If the triple integral of the function g is equivalent to the triple integral of the function w, is it the case that g=w?
0
votes
2answers
38 views

Find $\iiint_E sin^3 x+\tan y+ 6\hspace{1mm} dV$, where $V$ is region inside $x^2+y^2+z^2 = 1$

I guess that the integral of $\sin^3 x+\tan x$ part is zero, because i have seen many problems like these where the integral is over a symmetrical region and the functions are odd. But I want ...
2
votes
2answers
75 views

Why is continuous differentiability required?

I have two questions. My book proves that if $f:\mathbb{C}\rightarrow \mathbb{C}$ is a holomorphic function, then it satisfies the Cauchy-Riemann equations, and if we look at the function as $F: ...
3
votes
1answer
95 views

Finding multivariable limit

I would like to find the following limit $$\lim_{(x,y,z)\to(0,0,0)}\frac{x^3yz+xy^3z+xyz^3}{x^4+y^4+z^4}.$$ It looks like it would be zero since if we put $M=\max\{x,y,z\}$ and $m=\min\{x,y,z\}$, ...
5
votes
0answers
55 views

Symbolic manipulation inside integral

I'm an undergrad who has just completed the standard calculus sequence (1, 2, and multivariable). I've done well in the courses, however, things like the following, which is a derivation of kinetic ...
3
votes
2answers
98 views

$\Delta \vec{v}=0$ implies $\nabla\cdot \vec{v}=\nabla\times \vec{v}=0$?

\begin{align} \Delta\overrightarrow{v}&=\nabla(\nabla\cdot\overrightarrow{v})-\nabla\times(\nabla\times\overrightarrow{v})\\ ...
2
votes
2answers
103 views

A continuously differentiable function with vanishing determinant is non-injective?

(This question relates to my incomplete answer at http://math.stackexchange.com/a/892212/168832.) Is the following true (for all n)? "If $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ is continuously ...
2
votes
0answers
35 views

Proof of Second Partials Test

How does one rigorously prove the second partials test without firstly assuming that $D(a,b)=AC-B^2$ that states the following: $ A=\frac {\partial^{2}f(a,b)}{\partial x^{2}},B=$$\frac ...
0
votes
1answer
36 views
+50

Converting a slope field into a vector field

I have homework on slope fields where I have to graph a bunch and find the equillibrium solution, but instead of taking such a long time to graph them, I decided to use WolframAlpha. Sadly, there is ...
2
votes
2answers
28 views

Extract a variable from a formula

my maths a a bit rusty and I need to extract a variable from a formula. It's needed for a project about air quality in order to convert data from sensors to an index. The formula is : $$\left ...
0
votes
1answer
21 views

Prove that the gradient transforms as a vector under rotations

I have not been able to make the following problem: Consider that $f$ is a function of only two variables, $y$ and $z$. Show that the gradient: $$\nabla f=\left(\frac{\partial f}{\partial ...
0
votes
3answers
63 views

Area of the region: $\;x ≥ 0; \;−x\sqrt3 ≤ y ≤ x\sqrt3;\,\;(x−1)^2 + y^2 ≤ 1$.

Can anyone please explain how to set up the needed integral? I need to calculate the area of the region given by: $x ≥ 0,$ $-x\sqrt3 ≤ y ≤ x\sqrt3,$ $(x−1)^2 + y^2 ≤ 1$.
0
votes
1answer
26 views

Evaluating multivariable limit $\lim_{(x,y)\to(0,0)} \frac{x^3-y^4}{\sqrt{x^2+y^4+1}-1}$

If it exists, find the following limit: $$\lim_{(x,y)\to(0,0)} \frac{x^3-y^4}{\sqrt{x^2+y^4+1}-1}.$$ I tried the following: $$\begin{align*} \frac{x^3-y^4}{\sqrt{x^2+y^4+1}-1} = ...
1
vote
2answers
47 views

Show that the function $f(\textbf{x}) =|\textbf{x}| $ is continuous on $\mathbb{R}^n$

I can see this intuitively, but looking for a solid answer with reasoning. all ideas will be appreciated,
1
vote
2answers
231 views

Continuity of piecewise function

$$f(x,y) = \begin{cases} \dfrac{\sin(xy)}{xy} & \text{if $x y \ne 0$} \\ 1 & \text{if $xy=0$} \end{cases}$$ all ideas are appreciated i think this is non-continuous, i did by converting to ...
0
votes
1answer
25 views

Why is the derivative Df(p) defined to $ \in \Lambda^1 (\mathbb{R}^n) $, or how is it a 1-form?

I know that obviously the differential operator D would be a differential form through the word differential. But in Spivak Calculus on Manifolds he defines a k-form w $ \in \Lambda^k ...
2
votes
2answers
68 views

System of equations in Lagrange multiplier problem

Continuing from Confounding Lagrange multiplier problem: I'm having trouble solving the system of equations below arisen from a Lagrange multiplier problem where we are to optimize $f(x,y,z) = 4x^2 + ...
1
vote
2answers
91 views

Using Stokes theorem to integrate $\vec{F}=5y \vec{\imath} −5x \vec{\jmath} +4(y−x) \vec{k}$ over a circle

Find $\oint_C \vec{F} \cdot d \vec{r}$ where $C$ is a circle of radius $2$ in the plane $x+y+z=3$, centered at $(2,4,−3)$ and oriented clockwise when viewed from the origin, if $\vec{F}=5y ...
0
votes
1answer
64 views

Triple Integral Volume Question

The question asks for the triple integral of $e^{-(x^2+y^2+z^2)^{3/2}} dV$ where $D$ is a sphere of radius $4$. The answer that I came up with is $2(1-e^{-64})$. However, I am not confident in this ...
8
votes
1answer
52 views

Changing the order of integration without sketching?

When changing the order of double integrals, I have always relied on sketching the region. I have recently come across this example on MSE by @FelixMartin which seems to avoid visual-based reasoning, ...
0
votes
2answers
60 views

Multiplication and derivation of 3D matrix

I have $A(q)=\begin{bmatrix}q_1 &q_2 & q_3\\ 2q_1 &3q_2 & 4q_3\\ 2q_1 &3q_1 & 10\\ \end{bmatrix}\tag 1$ $ q= {\left(\begin{array}{c}q_1\\q_2\\q_3\\q_4\\q_5\\q_6 ...
1
vote
0answers
24 views

Showing that $f\in C'(\mathbb{R^2},\mathbb{R})$

Let $$f(x,y)=xy\int_{x^2-y^2}^{x^2+y^2}e^{\cos(xyt)}dt.$$ Prove that $f\in C'(\mathbb{R^2},\mathbb{R})$. I'm not exactly sure how to approach this problem. Here's what I've tried: First I ...
3
votes
2answers
39 views

Solving $4y^4 - 4x^4 + x + y = 0$ (equation system of partial derivates)

I need help solving the following equation system: $$ \frac{\partial}{\partial x} = 8xy + 4y^2 + \frac{y}{x^2 + y^2} = 0 $$ $$ \frac{\partial}{\partial y} = 8xy + 4x^2 - \frac{x}{x^2 + y^2} = 0 $$ ...
1
vote
2answers
28 views

Evaluating a polar double integral on the semi disc

The integral: $$\iint_D (x^2-y^2)\,dx\,dy$$ where $D$ is defined as: $$\{(x,y)\in \mathbb R^2 \mid x^2+y^2\le 1, x\ge 0\}$$ Context I have solved double integrals on quarter discs but this semi ...
1
vote
2answers
80 views

Find $ \int_0^2 \int_0^2\sqrt{5x^2+5y^2+8xy+1}\hspace{1mm}dy\hspace{1mm}dx$

I need the approximation to four decimals Not sure how to start or if a closed form solution exists All Ideas are appreciated
2
votes
1answer
43 views

verifying extrema found by Lagrange multipliers

This question was inspired by reading this problem: Prove the inequality $\frac 1a + \frac 1b +\frac 1c \ge \frac{a^3+b^3+c^3}{3} +\frac 74$ Suppose I have a function $f(x,y,z)$ with continuous ...
2
votes
2answers
43 views

Two and Three Variable Limit Questions

Find the following limits, if they exist. $$\lim_{x,y\rightarrow 0,0}\frac{x^2 + \sin^2 y}{\sqrt{x^2+y^2}}$$ I believe we're suppose to use the squeeze theorem on this first one above. Possibly ...
0
votes
2answers
20 views

Marginal density function question

The question and answer is shown but I don't fully understand the answer for part a. Could someone please explain to me why the integral setup for the marginal density function of y1 is from y1 to 1, ...
0
votes
1answer
67 views

Splitting Integral into Two Parts

This question might seem very simple, but I can't seem to figure it out. Suppose I have an integral over a square region. I was wondering in which case it would be incorrect to split the integral into ...
0
votes
1answer
25 views

Vectorial Calculus proof [closed]

Please help me to prove the following identity wherein $\phi$ is a scalar field and d$\vec l$ is the linear element: $$\int \nabla \phi \cdot d\vec l = \int d\phi$$ hopefully step by step. ...
0
votes
1answer
36 views

Deriving FTC from the generalized Stokes.

How do I derive the Fundamental Theorem of Calculus from the generalized Stokes theorem?
0
votes
1answer
67 views

Two methods of finding a function $f$ such that $Mdx+Ndy=0$ on the curves $f(x,y)=c$

this problem is from my class,i did one way and got one answer,professor did it in another way and got another answer.question is:Find $f(x,y)=constant$ where differential equation is ...
2
votes
1answer
40 views

direction limits and double limit

Let $f(x,y)$ be a function of two variables. What is the counterexample that there exists $A$ s.t. for all $\theta$, $$\lim_{r\to 0+}f(r\cos \theta,r\sin \theta)=A$$ but double limit $$ ...
0
votes
2answers
24 views

Finding a value R that maximizes the flux a vector field over half a sphere of radius R

Sorry for the bad title, couldn't think of a less convoluted way of writing it. I have to find $ R\in \mathbb{R}$ so that the flux of $$F(x,y,z) = (xz - x\cos(z), -yz +y\cos(z), -4 - (x^2 + y^2)) $$ ...
4
votes
1answer
41 views

Derivative: $f_x, f_y, f_{xy}$ of function - $f(x,y)$

Let's say $f(x,y) = x^2 + 2xy +y^2$ $f'_x = 2x + 2y$ $f'_y = 2y + 2x$ $f'_{xy} = 2x + 2y$ ? Am I right about the third?
2
votes
2answers
186 views

A limit two variables

How can I compute or prove that $\displaystyle\lim_{(x,y)\to(0,0)}\dfrac{\mathrm{e}^{xy}-1}{\sqrt{x^2+y^2}}=0$?
0
votes
1answer
56 views

Find the volume inside

Find the volume inside the torus $\rho=\sin\phi$. First of all how can $\rho=\sin\phi$ represent a torus? I can't even visualise that. All Ideas are welcome, this looks like a 'food for thought ...
0
votes
1answer
42 views

Find $\int_0^1 \int_{3x}^3 (x^2+y^2)\sqrt{9-y^2}\hspace{1mm}dy dx$ [closed]

You can use a calculator after simplification if its not possible by hand All Ideas will be appreciated Also If you could find $$\int_0^1 \int_{3x}^3 x(x^2+y^2)\sqrt{9-y^2}\hspace{1mm}dy dx$$ ...
2
votes
2answers
34 views

Maximum and minimum of $z=\frac{1+x-y}{\sqrt{1+x^2+y^2}}$

Find the maximum and minimum of the function $$z=\frac{1+x-y}{\sqrt{1+x^2+y^2}}$$ I have calculated $\frac{\partial z}{\partial x}=\frac{1+y^2+xy-x}{(1+x^2+y^2)^{\frac{3}{2}}}$ $\frac{\partial ...