Use this tag for questions about differential and integral calculus with more than one independent variable. Some related tags are (differential-geometry), (real-analysis), and (differential-equations).

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4
votes
0answers
107 views

closed form is exact in euclidean space

Question is to show that $d(f)=0$ for a $0$ form on $\mathbb{R}^n$ then $f$ is a constant function. See that $$0=df=\sum_i\frac{\partial f}{\partial x_i}dx_i$$ implies that $\frac{\partial ...
2
votes
1answer
14 views

Tangent space of manifold has two unit vectors orthogonal to tangent space of its boundary

I'm reading spivak calculus on manifolds and got stuck. Let M be a k-dimensional manifold with boundary in $\mathbb{R^{n}}$, and $M_{x}$ is the tangent space of M at x with dimension k, then $\partial ...
1
vote
1answer
33 views

How to interpret a mapping in $\mathbb{R}^{2}$

So I am trying to find the image of the circle $(x-1)^{2} + y^{2} = 1$ under the mapping $F$ defined by $$(u,v) = F(x,y) = \bigg(\frac{1}{2}(x+y), \frac{1}{2}(-x+y)\bigg)$$ Using computational ...
1
vote
2answers
51 views

Notation in Vector calculus, Stokes' theorem

I have a question regarding Stokes' theorem: $$\oint_c \vec{F} \, d\vec{r} = \iint_S \nabla \times \vec F({r}(u,v)) \cdot d\vec S = \iint_S \nabla \times \vec F({r}(u,v)) \cdot (r_u \times r_v)\, ...
2
votes
2answers
36 views

How to find cartesian coordinate of velocity of particle on the trajectory, $Ax^2 +2Bxy +Cy^2=1, A,B,C >0.$

Consider a particle with constant speed $|w|=w_o$ moving on trajectory $Ax^2 +2Bxy +Cy^2=1, A,B,C >0.$ Could anyone advise me how to express cartesian coordinates of $v$ in terms of $x$ and $y \ ?$ ...
1
vote
1answer
23 views

Counter Example Problem ( Two variable function ).

In the given situation we show that , either the statement is true or we find a counter example to prove it wrong , If $\lim_{y \to 0} f(0,y)=0$ , then , $\lim_{(x,y) \to (0,0)} f(x,y)=0$ I ...
1
vote
2answers
23 views

Limits (Three Variable function).

We're given : $f(x,y,z) = \dfrac{xyz}{x^{2}+y^{2}+z^{2}}$ , Also , it's given that $\lim_{(x,y,z) \to (0,0,0)} f(x,y,z)$ exists. We need to prove that $\lim_{(x,y,z) \to (0,0,0)} f(x,y,z) = 0$. ...
0
votes
0answers
24 views

Finding Limits of Integration: Developing Intution

I am having trouble finding limits of integration in Multivariable Calculus. My question is that is there a way to find these bounds without graphing. I'm just not able to understand how to find these ...
1
vote
1answer
37 views

Extending a smooth function of constant rank

Let's denote $\mathbb{H}^m = \{(x_1, \ldots, x_m) \in \mathbb{R}^m\ |\ x_m \geq 0\}$. For an open subset $U \subset \mathbb{H}^m$, a function $f : U \to \mathbb{R}^n$ is called smooth if it can be ...
1
vote
1answer
30 views

Integration By Parts on a Fourier Transform

I'm having trouble with the "An integration by parts in $x$ for the first summand...and the assumption that $\phi$ goes to $0$ as $|x|\to\infty$." I tried the integration by parts but ended up with ...
5
votes
0answers
51 views

Continuity ( Functions of 2 variables ).

Given , $$ f(x,y) = \begin{cases} \dfrac{xy^{3}}{x^{2}+y^{6}} & (x,y)\neq(0,0) \\ 0 & (x,y)=(0,0) \\ \end{cases} $$ We need to check whether the function is continuous at ...
2
votes
1answer
38 views

Non linear system of differential equations

Is there a specific name to the following type of non linear ODEs $\begin{array}{c} \dot{x}_1 &= c_1 \, x_2\, x_3 \\ \dot{x}_2 &= \, c_2 x_1 x_3 \\ \dot{x}_3 &= c_3 \, x_2 x_1 ...
4
votes
2answers
170 views

Minimum of an apparently harmless function of two variables

I would like to prove that the minimum of the function $$ f(x,y):=\frac{(1-\cos(\pi x))(1-\cos (\pi y))\sqrt{x^2+y^2}}{x^2 y^2 \sqrt{(1-\cos(\pi x))(2+\cos(\pi y))+(2+\cos(\pi x))(1-\cos(\pi y))}} $$ ...
2
votes
1answer
27 views

Finding the net outward flux of a sphere

Use the Divergence Theorem to compute the net outward flux of: $$ F = \langle x^2, y^2, z^2 \rangle $$ $S$ is the sphere: $$ \{(x,y,z): x^2 + y^2 + z^2 = 25\} $$ First, I took: $$ \nabla \cdot F ...
0
votes
1answer
30 views

How do you go about solving partial differential equations for finding critical points in general optimization problems?

I was reading about partial second derivative test for optimization problems and I came across the example here. I saw the equations have yielded four critical points, but I wasn't able to find those ...
0
votes
0answers
53 views

Partial derivative of recursive exponential $f(x) = \sum^{K_2}_{k_2=1}c_{k_2} \exp(-z^{(2)}_{k2})$ with respect to the deepest parameter

I was trying to take the derivative of the following equation (which can be depicted nicely in a tree like structure, look at the end of question for diagram): $$f(x) = f([x_1, ..., x_{N_p}])= ...
0
votes
2answers
28 views

Is the funtion $f(x,y)=\frac {x^2y^2}{x^2y^2 + (y-x)^2}$ when $(x,y)\neq (0,0)$ and $f((0,0))=0$ continuous at $(0,0)$ and is this differentiable?

Is the function $$f(x,y)=\begin{cases}\frac {x^2y^2}{x^2y^2 + (y-x)^2} & \text{ , when } (x,y)\not=(0,0)\\0&\text{ , when }(x,y)=(0,0)\end{cases}$$ continuous at $(0,0)$ is this ...
0
votes
1answer
34 views

Conditions on a linear system of ODEs

Let $x:[0,T]\to\mathbb{R}^n$ and $y:[0,T]\to\mathbb{R}^n$ be solutions to an $n\times n$ system of linear ODEs. That is, $$\frac{dx}{dt}=A(t)x+b(t) \mbox{ and } \frac{dy}{dt}=A(t)y+b(t) \mbox{ for } ...
1
vote
1answer
64 views

Finding $\iint_S {z \:ds}$ for some $S$

$$\iint_S {z \:ds}$$ In this double integral above, $S$ is the part of a sphere, $x^2+y^2+z^2=1$, which lies above the cone, $z=\sqrt{x^2+y^2}$. How can I calculate the above double integral. Can ...
3
votes
1answer
68 views

Using Stokes' Theorem Finding $\int_C{F\bullet dr}$

Suppose that $C$ is the intersection of $z=2x+5y$ and $x^2+y^2=1$ which is oriented counterclockwise when viewed from above. Now let $$F=\langle \sin{x}+y, \sin{y}+z, \sin{z}+x \rangle$$ How can I ...
0
votes
1answer
52 views

Proper definition use in Stoke's theorem

Let the curve C be a piecewise smooth and simple closed curve enclosing a region, D. Some sources asserts Stoke's theorem to be: $$\oint_{C} F.dr = \iint_{R}\nabla \times FdS$$ Whereas, some claims ...
1
vote
3answers
60 views

How to sketch a surface in a three-dimensional space?

I was asked to hand sketch the surface defined by $$x^2+y^2-z^2=1$$ How could I do that? I find it particularly hard to draw graph in three-dimension, could you give me some advice?
0
votes
1answer
38 views

differential forms question. [closed]

Let $ f: \mathbb{R^3} \to \mathbb{R}$ be the function $f(x, y, z) = x^2 + y^2 + z^2$ and let $F : \mathbb{R^2} \to \mathbb{R^3}$ be the map $$F(u,v)= \big( ...
0
votes
1answer
43 views

Differentiability of $\frac{x^2y^2}{x^2+y^4}$ at $(0,0)$ [closed]

Given function, $f$ defined: $f(x,y)=\frac{x^2y^2}{x^2+y^4}$ if $(x,y)\ne (0,0)$ and $f(0,0)=0$ Prove that $f(x,y)$ is not differentiable at $(0,0).$
1
vote
0answers
64 views

Path continuous but not continuous [closed]

Find an example of function $f : \mathbb{R}^2 \to \mathbb{R}$ such that $f$ is continuous along every path but $f$ is not continuous.
0
votes
1answer
21 views

Finding the derivative of a multivariable function

Suppose $f: \mathbb{R}^n \to \mathbb{R}$ is a differentiable function. Then we can write the derivative of $f$ as a $1 \times n$ row matrix of partial derivatives of $f$ ,i,e, ...
1
vote
2answers
66 views

how to differentiate $y(x) =exp(ax)$ twice

I'm quite confused with this differentiation: Suppose $x$ is a $m \times 1$ column vector, $a$ is a $1 \times m$ vector, I want to differentiate $\exp(ax)$ a few times. I think the first derivative ...
2
votes
2answers
46 views

does simply connectedness require connectedness?

My question consists of two parts. $1)$ suppose domain $D=\{(x,y)\in\mathbb R^2~|~xy>0\}$ is given. Now that is first quadrant and third quadrant with exclusion of $x$ and $y$ axis. We can easily ...
0
votes
1answer
19 views

Relationship between two variables with min and max value (please read inside) [closed]

Hi and sorry for the bad title. I do some programming for games and often run into the following practical problem: I have two values that run within certain limits, let's call them xMin, xMax, yMin ...
1
vote
1answer
23 views

$F = \langle yz-2xy^2, axz-2x^2y+z, xy+y \rangle$ [duplicate]

Given that: $$F = \langle yz-2xy^2, axz-2x^2y+z, xy+y \rangle$$ in which $a$ is some constant. Now, for what $a$ would make the vector field of $F$ conservative? How can we find an $f$ with $\nabla ...
0
votes
0answers
23 views

How to find out if point is local Maximizer or local Minimizer ? Lagrangian is given

The Lagrangian is: $L(x,\lambda) = x_1x_2-2x_1-\lambda (x_1^2-x_2^2)$ Taking the derivatives and setting it equal to zero gives: $x_2-2\lambda x_1-2=0$ $x_1+2\lambda x_2=0$ $x_1^2-x_2^2=0$ The ...
0
votes
4answers
49 views

Length of Spiral in a plane [closed]

Problem Take a positive constant real number $c$. Draw a rough sketch and find the length of the spiral in the plane given by $(x(t),y(t))=(e^{-ct}\cos(t),e^{-ct}\sin(t))$ for $0\leq t<\infty$. ...
0
votes
2answers
142 views

Math Subject GRE 1268 Problem 64 Flux of Vector Field

What is the value of the flux of the vector field F, defined on $R^3$ by $F(x,y,z) = (x,y,z)$ through the surface $z=\sqrt{1-x^2-y^2}$ oriented with upward-pointing normal vector field? ...
2
votes
5answers
42 views

Finding a general solution to a differential equation, using the integration factor method

Use the method of integrating factor to solve the linear ODE $$ y' + 2xy = e^{−x^2}.$$ And verify your answer I can solve the ODE as a linear equation (mulitply both sides, subsititute, reverse ...
1
vote
1answer
51 views

Multivariable Calculus: Manifolds

Problem Let $M$ be the set of all points $(x,y) \in \mathbb{R}^2$ satisfying the equation $$xy^3 + \frac{x^4}{4} + \frac{y^4}{4} = 1 $$ Prove that $M$ is a manifold. What is the dimension of $M$? ...
1
vote
1answer
43 views

Good reference on higher dimensional derivatives?

I've spent several months now periodically scouring the internet for a comprehensive overview of an introduction to higher dimensional derivatives. I've already read baby Rudin's section on the ...
1
vote
1answer
39 views

Surface integral on unit circle

Let $S$ be the unit sphere in $\mathbb{R}^3$ and write $F(x)=\nabla V(x)$ where $V(x)=1/|x|$ Evaluate $$\iint_S F\cdot n dS$$ Without using divergence theorem, we can evaluate it straightforwardly, ...
2
votes
1answer
25 views

Let $f$ be differentiable at every point of some open ball $B(a)$ in $\mathbb R^n$ and $f(x)\le f(a) , \forall x \in B(a)$ , then prove $D_k f(a)=0$.

If $f:\mathbb R^n \to \mathbb R$ is a function differentiable at every point of some open ball $B(a)$ with center $a\in \mathbb R^n$ and $f(x)\le f(a) , \forall x \in B(a)$ , then how to show that all ...
0
votes
2answers
19 views

$f'(x;y)=0$ for every $x$ in an open convex set and for every vector $y$ ; then to show $f$ is constant on $S$

Let $f:\mathbb R^n \to \mathbb R$ be a map , $S$ be an open convex set in $\mathbb R^n$ such that for every $x \in S$ and $y \in \mathbb R^n$ , $f'(x;y)$ exists and equals $0$ ; then how to show that ...
1
vote
3answers
55 views

Substitution to solve an initial value problem

By using the substitution $y(x) = v(x)x$, how can I solve the initial value problem $$ \frac{dy}{dx} = \frac{x^2+y^2}{xy - x^2},\quad y(1)=1 $$ And also keep my answer in the form $g(x,y)= 4e^{-1} ...
1
vote
1answer
32 views

Diffeomorphism between Euclidean space

How does one show that if $f:U\rightarrow V$ is a diffeomorphism between open sets $U\subset\mathbb{R}^m$ and $V\subset\mathbb{R}^n$ then $m=n$? Here is some working: For $u\in U$ let $v=f(u)\in V$. ...
1
vote
0answers
37 views

Application of Stoke's Theorem

Edit: I think I misunderstood the problem. Upon reading my textbook again, I think what they mean by $F(x,y,z)=<yz,2xz,e^{xy}>$ ; C is the circle $x^2+y^2=16, z=5$ is just literally a ...
0
votes
1answer
36 views

Stokes' Theorem - The normal vector

Stokes' theorem says: $$\oint_cFdr = \int\int_S curl F dS = \int\int_S curl F \cdot n \, dS$$ Where $F$ is a vector field on $\mathbb{R}^3$. My question is what do I take $n$ to be? If we ...
1
vote
1answer
45 views

Absolute Min and Max of $f(x, y)=x^2+4y^2-2x^2y+4$ Using Partial Derivatives

Consider this problem: Find the absolute minimum and absolute maximum of $f(x, y)=x^2+4y^2-2x^2y+4$ on the rectangle given by $-1\leq x\leq1$ and $-1\leq y\leq1$ I solved this problem using ...
0
votes
2answers
42 views

If a vector field is conservative then is it path independent?

I am studying vector calculus and I am having trouble with the idea of path independence. Is it necessarily true that if $F=(P,Q)$ (a vector field in $\Bbb R^2$) is conservative, then $\oint \limits ...
2
votes
2answers
92 views

Vector Field Conceptual Question

Given that: $$F = \langle yz-2xy^2, axz-2x^2y+z, xy+y \rangle$$ in which $a$ is some constant. Now, for what $a$ would make the vector field of $F$ conservative? Why is there only one, or are there ...
1
vote
0answers
47 views

Under what conditions is this true: $\lim_{r \to 0} \frac{1}{r} \int_{0}^{2\pi} f(r,x) dx = 2\pi f(0,0)$

I will like to know under what hypothesis the following is true, and maybe a sketch of the proof. I saw it in a solution of an exercise. In this exercise, $f$ was harmonic, but I don't know if that is ...
1
vote
2answers
84 views

Evaluating Line Integrals using Green's Theorem

I am currently learning about Green's Theorem, Curl and Divergence, and I came across a problem: Given a two dimensional vector field: $$ F=\langle e^{\sin{x}}+y^2, x^2+y^2 \rangle$$ And then I am ...
1
vote
1answer
26 views

Does given point satisfy FONC?

minimize $4x_1^2+2x_2^2-4x_1x_2-8x_2$ subject to $x_1+x_2\leq 4$ Does the point $(2,2)$ satisfy the FONC for a local minimizer? The gradient of the objective function is $\nabla f = ...
1
vote
2answers
17 views

Div$f$ is invariant under an orthogonal change of coordinates

Let $f: \mathbb{R^n} \to \mathbb{R^n}$ and $Df$ exists. I need to show that div$f$ is invariant under an orthogonal change of coordinates. Let $T:\mathbb{R^n} \to \mathbb{R^n}$ be an orthogonal ...