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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6 views

How to find critical points in a cubic function in two variables?

Given a cubic function $f$ in two variables $x$ and $y$ $$ f(x,y)=\sum_{i=0}^3 \sum_{j=0}^3 k_{i,j}x^i y^j, $$ I would like to find the points ($x,y$ pairs) where $\nabla f = \mathbf{0}$. Since $f$ ...
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1answer
15 views

Trying to find a scalar potential for a vector field whose domain is not connected or to prove it is not conservative

Let $U=\mathbb{R}^2-\lbrace(x,y)|xy+1=0 \rbrace$ and let $F:U \to \mathbb{R}^2$ be defined as $F(x,y)=(\frac{1-y^2}{(1+xy)^2},\frac{1-x^2}{(1+xy)^2})$ I am trying to find a scalar potential of $F$ in ...
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0answers
7 views

Flux integral over a parabolic cylinder

Evaluate $\int\int_S \textbf{F}\cdot\textbf{n} dS $ where $\textbf{F}=(z^2-x)\textbf{i}-xy\textbf{j}+3z\textbf{k}$ and S is the surface region bounded by $z = 4-y^2, x=0, x=3$ and the x-y plane with ...
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0answers
16 views

global inverse function theorem

Consider a continously differentiable non-constant function $f:\mathbb{R}^2\to\mathbb{R}$. Define $$ K=\{x\in\mathbb{R}^2|f(x)=0\}. $$ I wish to know whether there is a continuously differentiable ...
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1answer
28 views

Representing a triple integral in a different order of integration

I am given with the following question: A) $V_1 = \{ x^2 + y^2 \leq 4 , 0\leq z\leq 3 \sqrt{x^2 + y^2 } , x\geq 0 \} $ , and I need to represent the triple integral $\int \int \int_{V_1} f(z) dxdydz$ ...
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1answer
28 views

Calculate the integral $\iint_D (y^2-x^2)^{xy} (x^2+y^2)dxdy$ on a certain region

Let $D$ be the region that's bounded by $xy=a, xy=b, y^2-x^2=1, y=x$ in the first quadrant. Calculate the integral $\iint_D(y^2-x^2)^{xy}(x^2+y^2)dxdy$. Firstly, I was able to show that the boundary ...
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1answer
26 views

Converse of a theorem: If the curl of a vector field is not zero does it implies it is not conservative?

I have the following theorem: If $F$ is a vector field defined in a simply-connected open set, whose coordinate functions have continuous partial derivatives and $curl(F)=0$, then $F$ is ...
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7answers
52k views

What is Jacobian Matrix?

What is Jacobian matrix? What are its applications? What is its physical and geometrical meaning? Can someone explain with examples? Thanks! :)
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1answer
28 views

Partial Derivative and Differentiability

I need help for the following question. Do the partial derivatives of the function $f(x, y)=min(|x|,|y|)$ exist and what are they? Also, I don't think the function is differentiable at $(0,0)$ but ...
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0answers
15 views

Is $\sigma_u \times \sigma_v \neq \vec{0}$ essential for $\int_{\alpha} \vec{F} \cdot d\vec{r} = \iint_\sigma \mathrm{curl}\,\vec{F} \cdot d\vec{S}$?

I think I have proved the following version of Stokes' theorem: Teorem 1: Let $\beta: [0,4] \to \mathbb{R}^2$ be the curve given by \begin{equation} \beta(t) = \begin{cases} (t,0) & \mbox{if ...
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1answer
41 views

How can I calculate $\int_1^2 \int_{\sqrt{x}}^x \sin\left(\frac{\pi x}{2y}\right) \,dy \, dx$?

Good night i have problem solving this integral. $$\int_1^2 \int_{\sqrt{x}}^x \sin\left(\frac{\pi x}{2y}\right) \,dy \, dx$$ I make the area of integration, but i cannot solve the integrat, i don't ...
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1answer
32 views

Jacobian determinant

I understood that when using substitution for multiple variables, the Jacobian determinant should be added. \begin{vmatrix} \frac{\partial f_1}{\partial v_1} & \frac{\partial f_1}{\partial v_2} ...
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1answer
15 views

Show that f(x,y)=min(|x|,|y|) is not diferentiable at 0. [on hold]

I proved that the partial derivatives at 0 exists and equal 0. I do not know how to proceed.
2
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1answer
35 views

Problem solving $\int_{1}^{2}\int_{x}^{2x}\int_{\sqrt{1-x^{2}-y^{2}}}^{\sqrt{2xy}}\frac{zdzdydx}{x^{2}+y^{2}+z^{2}} $

Good night, i have a problem solving this integral: $$\int_{1}^{2}\int_{x}^{2x}\int_{\sqrt{1-x^{2}-y^{2}}}^{\sqrt{2xy}}\frac{zdzdydx}{x^{2}+y^{2}+z^{2}}$$ I think make a change to spherical ...
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0answers
11 views

Determining positivity of a blackbox multivariate polynomial

Is there a way to check the positivity (or non-negativity) of a multivariate polynomial $f: \mathbb{R}^n \to \mathbb{R}$, of a given degree $d$, by querying the value of $f$ at finitely many points?
2
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2answers
913 views

What does it mean by piecewise smooth boundary?

I will be highly obliged if anyone can give me any reference where i can get the definition of domain (in $\mathbb{R^n}$) with piecewise smooth boundary. My question is when a domain in ...
0
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1answer
12 views

Triple Integral of 6xy dV, why is lower bound of z 0 instead of 1?

In the Stewart Calculus Textbook 7th Edition, problem 13 in chapter 15.7 states: "Evaluate the triple integral: Triple Integral within E of 6xy dV, where E lies under the plane z=1+x+y and above the ...
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1answer
25 views

A specific case of quadratic forms

I have a quadric as follows: $$ax^2+by^2+bz^2+yz=0.$$ I am curious to know which shapes in $\mathbb{R}^3$ this equation describes for different value of $a$ and $b$?
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0answers
18 views

Linear algebra: Solving for the coefficients on vectors

I am solving the following system: $$ -\frac{1}{r^2}\begin{bmatrix}\sqrt{\mu}\cos(\theta)\\ \sin(\theta) \end{bmatrix}= ...
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1answer
26 views

Integrating $\iint_D x^2y\log(y) + 2x^2y \textrm{d}x\textrm{d}y$, where $D$ is the region bounded by $e^x \leq y \leq e^{2x}, \; 1 \leq x^2y \leq 2$

I need the calculate the following integral: $\iint_D x^2y\log(y) + 2x^2y \;\textrm{d}x\;\textrm{d}y$, where $D$ is the region bounded by $e^x \leq y \leq e^{2x}, \; 1 \leq x^2y \leq 2$ What is ...
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0answers
16 views

Do regions of integration cover *all* possible regions?

Now, I'm not real concerned about the integration itself but more about integration regions. Is it true that all possible regions in two-space are covered by the following setup: $$\int^a_b ...
2
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2answers
46 views

What is the difference between the gradient and the gradient vector?

Here is the where my confusion starts.... When they are speaking of curves, they are using the terms "gradient" and "slope" for slope. But when it comes to surfaces, it was stated that the gradient ...
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1answer
50 views

Issue with substituting a new variable

I have a function of the form $$u(x,t)=\int_{0}^{t}{\frac{u_0\,x\exp[-h\tau-(x^2/4k\tau)]}{2\sqrt{\pi k\tau^3}}}d\tau$$ Now substituting $\eta=\frac{x}{2\sqrt{k\tau}}$ in the above equation, I get ...
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0answers
23 views

Linear Transformation from alpha to beta [on hold]

Hello I am in my Calculus 4 class and I am studying for the final and one thing I've not ever been able to understand is how to do that matrix representation. so I'm working on the practice final ...
0
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2answers
392 views

Using Lagrange for finding Marshallian Demand

I want to find the marshallian demand function for the user function $u(x_1,x_2) = x_1^ax_2^{1-a}$ where $a \in (0,1)$. This is what I have so far: $$L = x_1^ax_2^{1-a} - \lambda(p_1x_1 + p_2x_2 - ...
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0answers
15 views

Concerning the notation $\chi (U)$ in one of the hipotesis for some propierties of curl and divergence

I have the following excercise: Let $U \subset \mathbb{R}^3$ be open, $X \in \chi (U)$ and $f \in C^{\infty}(U)$, prove the following: $$curl(\nabla f)=0 \\ div(curl(X))=0 \\ curl(f.X)= ...
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0answers
22 views

What is the general Taylor Expansion for the following function of a function.

guys. I am stuck with a general form of Taylor Expansion of following function, which is defined as a function of a function: ...
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1answer
95 views

Isothermal parameterization, Inverse of the Gauss Map

This problem is from Do Carmo's Differential Geometry of Curves and Surfaces. It is question 13 from chapter 3.5, to be specific. Suppose that S is a minimal surface without any umbilical points ...
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1answer
56 views

Apparent violation of fundamental theorem of ODEs, how to resolve?

Consider, in the $(x, y)$-plane, the family of curves given by $y = (x - c)^3$, for the various possible values of the number $c$. Denote by $v$ the unit vector field everywhere tangent to this family ...
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0answers
29 views

Problem finding the tangent plane and the normal line of an surface [on hold]

Good night, I have a serious problem when I try to find a tangent plane for the following surface at the point $P$: $$x^{2}+y^{2}+z^{2}=6, \hspace{4mm} P=(-1,-2,3).$$ I make this: $\nabla ...
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0answers
75 views

Expectation or Integration of the normal cdf

Can any one help me how to solve this pronbelm? I have a random variable $W$, i.e., $$W=\Phi(X)^k\Phi(-X)^m=P(Z\le X)^kP(Z \ge X)^m,$$ $X$ is Normal($\mu$,1), $Z \text{ is Normal(0,1)}$, and $k$ ...
0
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1answer
27 views

Problem solving a partial derivative with a integral. [on hold]

Good night, i have a serious problem solving this partial derivative: $f(x,y)=\int_{y}^{x}e^{t^{2}}dt$ I don't know how i can start this, please give me a help, don't do it the exercise, only explain ...
2
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1answer
42 views

Magnetic field by current in an infinite cylinder

Let $V\subset\mathbb{R}^3$ be an infinitely high solid cylinder of radius $R$, with its axis coinciding with the $z$ axis, entirely enclosed by the cylinder's lateral surface. Then, for any constant ...
3
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2answers
51 views

Find The Volume of the solid in the first octant , limit by: $ x^2+y^2=4 $ and $z+y=3$

Find the volume of the solid in the first octant , limit by: $ x^2+y^2=4 $ and $z+y=3$. $x$ and $y$ range from $0$ to $2$. $$\int_0^2 \int_0^2 y-3 \,dy\,dx $$ is correct?
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1answer
71 views

f continuously differentiable implies f is Lipschitz on compact subsets

It is a more general form of the question here, only here $U$ is not a convex set but an open and connected subset of $\mathbb{R}^n$. I need to show that $f$ is $M$ Lipschitz on any compact $K \subset ...
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1answer
38 views
+50

Surjectivity of derivative of a vector valued function

Let $f:\mathbb R^3\to \mathbb R^3$ be a function such that $f(x,y,z)=f(x+y,0,x+z)$ for all $(x,y,z)\in \mathbb R^3$. I want to prove that $f^{'}(x)$ can never be onto for all point $x\in \mathbb R^3$ ...
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1answer
20 views

Find the surface area of the shape formed by the boundary of $\frac{z^2}{4}=\frac{x^2}{2}+\frac{y^2}{4},z=2x+4y, z\geq 0$

$$\frac{z^2}{4}=\frac{x^2}{2}+\frac{y^2}{4},z=2x+4y, z\geq 0$$ I know that this is a cone that is cut by a plane, but I do not know how to find the projection onto $xOy$. I need this because then I ...
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3answers
444 views

How to prove mathematically that two planes parallel to a third plane are parallel

Without relying on geometrical intuition and purely using vector calculus, how do we show that two planes parallel to a third plane are parallel? I assume three dimensional space.
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2answers
50 views

Local extrema and minima of the multivariable function $f(x,y) = x^2y+y^2+xy$

Let $f(x,y) = x^2y+y^2+xy$ be a function, I want to find its local extrema an minima. I easily find that $f$ has 2 critical points: $(x,y)=(0,0)$ and $(x,y) = (-1,0)$. In order to find its local ...
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1answer
628 views

Parametrization of the intersection of a cone and plane.

EDITED with new progress updates. As the title states, I'm trying to parametrize the intersection of a cone and a plane. The equations are: $z^2 = 2x^2+2y^2$ and $2x+y+3z=4\implies ...
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1answer
21 views

let F be velocity vector field of fluid on $R^3$ defined by F(x,y,z)=-yi+xj.

$\newcommand{\Reals}{\mathbf{R}}\newcommand{\Vec}[1]{\mathbf{#1}}$Let $F$ be velocity vector field of fluid on $\Reals^3$ defined by $F(x,y,z) = -y\Vec{i} + x\Vec{j}$. (A) Show that $F$ is ...
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1answer
30 views

Word Problem Lagrange Method

I am studying for my exams and got very very stuck at a word problem on the Lagrange Methods, my biggest difficulty is to properly identify the function to be maximized (in this case) and so its ...
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1answer
19 views

Trying to find the square with minimum area inscribed in a Square of side L

A square has side length of L. Using the lagrange's multipliers, show that all squares inscribed in the square of side length of L, the square with minimum area has a side length of (sqrt(2)/2)L. I ...
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0answers
74 views
+50

Intuitively what is the second directional derivative?

I'm thinking that the second directional derivative, if both dd's are evaluated in the same direction, will just give you the concavity (the second scalar derivative) in that direction. Is that ...
2
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2answers
27 views

$\lim_{(x,y)\to (0,0)} \frac{\sin(x^2+y^2)}{x^2+y^2}$

I must admit that I've forgotten how to do multivariable limits. Nevertheless I need to know whether the following exists: $$\lim_{(x,y)\to (0,0)} \frac{\sin(x^2+y^2)}{x^2+y^2}$$ Would it be as ...
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1answer
41 views

Finding Extremas of $|x|$.

I'm trying to find the extrema of$\mod(x)$ but I'm not being able to do so. My attempt: $f(x, y) = |x|$ $f_{xx} = 0, f_{yy} = 0, f_{xy} = 0.$ So, $D(x, y) = 0$. And second derivative test isn't ...
7
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1answer
207 views

Volume of a sphere by “adding” half-spheres of lower dimension

I'm wondering about different ways to compute the volume of an $n$-sphere. Please see the wikipedia page for one method to compute the volume via hyperspherical coordinates: ...
3
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2answers
19 views

Integrating Line Segment and Path

Just wondering if anyone can give some assistance. I'm stuck on an exam question: $\ \vec F (x,y,z) = (6xy + 4xz)\vec i + (3x^2 + 2yz)\vec j + (2x^2 + y^2)\vec k , x,y,z ∈\Bbb R.$ Evaluate $$\ ...
2
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1answer
53 views

$\iint_{\mathbb R^2}\sqrt{\frac{x^2}{a^2}+\frac{x^2}{b^2}}e^{-\frac{x^2}{a^2}+\frac{y^2}{b^2}}dxdy$

$$\iint_{\mathbb R^2}\sqrt{\frac{x^2}{a^2}+\frac{y^2}{b^2}}\,e^{-\left(\frac{x^2}{a^2}+\frac{y^2}{b^2}\right)}\,dx\,dy$$ Basically I have done problems similar to this, using the theorem that if ...
0
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3answers
20 views

How to approach $\phi: \Bbb R^3 \to \Bbb R$ such that $\vec F = \nabla\phi.$

I'm looking through exam papers and I'm lost on what to do when asked to find a function $\phi: \Bbb R^3 \to \Bbb R $ such that $ \vec F = \nabla\phi.$ An example of a question I'm looking at is as ...