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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1answer
25 views

To find the Partial derivatives of x and y

Given $f(x,y) = \int_x^y \! g(t) \, \mathrm{d}t.$ g is continuous for all t. I need to find partial w.r.t x and y Since no function g is given, then i won't be able to integrate and compute ...
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2answers
36 views

How to show that $f(x,y) = |x| + |y|$ is continous at origin

How to show that $f(x,y) = |x| + |y|$ is continous at origin. CLearly it goes to 0 , but how do i prove it? Thanks
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0answers
22 views

Positiveness of a real valued function of $n$ variables??

Let $\{a_1,...,a_n\}$ be a set of $n$ non-negative parameters, we define $x^*=(x_i^*)_i$ as the $n$ dimensional vector with components: $$x_i^*=\frac{a_i^2}{\sum_j a_j^2}$$ Let $F:\Delta\to ...
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2answers
33 views

Understanding reasoning behind certain bounds in double integration.

So I am looking at the following question Find the volume of the ice-cream cone shape given by the region bounded between the upper half of the sphere $x^{2}+y^{2}+z^{2}= 16$ and the cone $z=\frac ...
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0answers
27 views

How to show a degenerate critical point is a saddle point.

The function $f:\mathbb{R}^2\rightarrow\mathbb{R}$ is given by $$f(x,y) = (x-y^2)(x-2y^2)$$ and I have found that the origin is a degenerate critical point (i.e. Hessian=0). How can I now show that ...
0
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1answer
27 views

Writing the change in a $\mathbb{R}^2 \rightarrow \mathbb{R}$ function in terms of its mixed partial derivatives

$f$ is defined in open set $S \subset \mathbb{R}^2$ and at all points in $S$ partial derivatives $D_{1}f$ and $D_{21}f$ exist. Suppose $Q \subset S$ is a closed rectangle with sides parallel to the ...
1
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1answer
58 views

Lagrange Multiplier Method: Why is the Langragian function defined as $f(x,y)+\lambda \cdot g(x,y)$?

Edit: As AlexR points out in this comment, there is no mathematical reason behind defining the Lagrangian, except because it makes the Lagrange Multiplier Method easier to memorize. I find this ...
2
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2answers
78 views

Finding a Surface Integral for the Vector Field: $F=\langle xz,x^2+y^2,y \rangle$

I need help calculating $$\iint_S F\cdot ds$$ where $$F=\langle xz,x^2+y^2,y \rangle$$ and $$S=\left\{(x,y,z)\mid x^2+y^2+z^2=25 ,y\ge0\right\}$$ oriented in the positive $y$ direction. My Thoughts: ...
2
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1answer
63 views

Calculating the Flux of $F$ over $S$

I need help calculating $$\iint_S F\cdot ds$$ where $F=\langle z,y,x \rangle$ and $$S=\left\{(x,y,z)\mid \frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1\right\}$$ and is oriented outwards. Would ...
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0answers
30 views

A doubt regarding the Jacobian in multivariable calculus

Let $f:\mathbb R^n\to\mathbb R^n$ be continuously differentiable on some open set containing $a\in\mathbb R^n$. Suppose $\det Jf(a)\neq0$. This is the hypothesis. Actually this is part of the ...
2
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0answers
31 views

Show that a subset is a submanifold of $ \mathbb{R}^4$

Given the functions $f_i : \mathbb{R}^4 \to \mathbb{R}, i = 1, 2, 3$ with $f_1(x) = x_1x_3 - x_2^2$ $f_2(x) = x_2x_4 - x_3^2$ $f_3(x) = x_1x_4 - x_2x_3$ show that $M:= \{ x \in ...
2
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2answers
47 views

If $f:\mathbb R^n\to\mathbb R$ is twice continuously differentiable, then $\nabla f$ is Lipschitz continuous

Let $f\in C^2(\mathbb R^n)$. I've read that since $f$ is twice continuously differentiable, $\nabla f$ is Lipschitz continuous. Is that really true? By the mean-value theorem, $$\left\|\nabla ...
2
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2answers
64 views

calculate $ \lim_{(x,y)\to (0,0)} \frac{e^{\frac{-1}{x^2+y^2}}}{x^4+y^4} $

calculate the limit: $$\lim_{(x,y)\to (0,0)} \frac{e^{\frac{-1}{x^2+y^2}}}{x^4+y^4} $$ I have tried to use polar coordinates, I also tried to show there is no limit, but I'm pretty sure now the ...
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1answer
24 views

Vector and parametric form of the equation of lines tangent to a surface

If we plot the graph of any $z=f(x,y)$ we will get a surface. If we take the partial derivatives at $(x_0,y_0)$ we will have two partial derivatives $f_x$ and $f_y$. The equation of the tangent lines ...
1
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1answer
21 views

Double Integral in Polar-Coordinates, Finding the constraint on r

Sketch the region of integration and then compute the following double integral: $$\int_0^1 \int_y^{\sqrt{2-y^2}}(x+y)dxdy$$ https://www.desmos.com/calculator/k5uz2mr2ml Then I characterized ...
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0answers
14 views

Identifying and classifying critical points of a general $\mathbb{R}^2\rightarrow\mathbb{R}$ quadratic function with only second degree terms.

Given $f(x,y) = ax^2 + bxy + cy^2.$ I am trying to find the critical points of $f$ and classify each as rel min, rel max, saddle, or degenerate. I understand how to use the determinant of the Hessian ...
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2answers
38 views

Definition of integration in *Calculus on Manifolds*

On page 65 of Calculus on Manifolds, right after defining a partition of unity $\Phi$, it states: If $\Phi$ is subordinate to $\mathcal{O}$, $f:A \to \mathbb{R}$ is bounded in an open set ...
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1answer
44 views

The explicit expression for integral of forms

Could anyone please help me with the following three questions? They are simple questions, but I am confused. With a $2$-form $F=\frac{1}{2}F_{\mu\nu}dx^{\mu}\wedge dx^{\nu}$ in 4 dimension, what is ...
0
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1answer
73 views

Limit Question, Multivariable Function

Its given that $\lim_{(x,y)\to (0,0)}f(x,y)=a $ exists and that $\lim_{x\to 0}\lim_{y\to0} f(x,y)=b$ also exists. Prove that a=b. We've always thought that the first limit exists only if the ...
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0answers
31 views

Bump function which $=1$ exactly on some compact set

Given an open set $O \subset \mathbb{R}^n$, and a compact set $K \subset O$, there exists a $C^\infty$ function $\varphi: \mathbb{R}^n \to [0,1]$ which is constantly $1$ on $K$ and is zero outside ...
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0answers
99 views
+50

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
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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 ...
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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
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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 $y \ ?$ ...
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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
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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 $\lim_{(x,y,z) \to (0,0,0)} f(x,y,z) = 0$. ...
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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
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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
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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 ...
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0answers
50 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
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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
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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
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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 ...
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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}])= ...
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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
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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
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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
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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 ...
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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?
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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 $(0,0).$
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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
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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 ...