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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54 views

How can show the following function has a global maximum?

I want to show that $$f(\alpha_1, \alpha_2, \ldots, \alpha_m)=\frac{\prod_{i=1}^{m}\alpha_i^{b_i}}{(1+\sum_{i=1}^{m}\alpha_i)^{\sum_{i=1}^{m+1}b_i}\prod_{j=1}^{2}[\sum_{i=1}^{m}\alpha_ic_i+d_j]^{a_j}}~...
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1answer
32 views

Degree of Jacobian of homogeneous polynomials

What is the degree of the Jacobian (as a polynomial) of 3 homogeneous polynomials in 3 variables of degrees say $m_1, m_2$ and $m_3$ ? I don't know how to prove that it is independent. In my case the ...
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1answer
27 views

Trouble with Triple integral in spherical coordinates

What is the form of triple integral to compute the volume bounded by $z=x^2+y^2$ and $z=9$ ? Also, change it to spherical coordinate?
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2answers
52 views

Implicit function theorem exercise problem

By eliminating $u$ from the equations, $x=u+v$ and $y=uv^2$, we get that $F(x,y,v)=0$. Where $v$ implicitly defines $x$ as function of $y$ ($v=h(x,y)$). Prove that $\dfrac{\partial h}{\partial x}=\...
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1answer
47 views

Why is a curve parameterized by arc length necessarily a unit speed curve? [duplicate]

I apologize if this is trivial but I have not been able to figure it out. For a curve $\sigma(t)$, I have a definition for arc length: $$s(t)=\int_{t_0}^t |\sigma'(t)|dt$$ We reparameterize a curve ...
1
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1answer
38 views

Equivalent form of a vector area of a surface

I am interested in showing that the vector area $$\int_{\mathcal{S}}da$$ can be equivalently given by $$\int_{\mathcal{S}} da = \frac{1}{2}\oint(r \times dI).$$ I am mostly interested in getting a ...
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0answers
11 views

Formal proof on continuity of multivariable function.

Let $B(o,r) = ${$(x,y)\in \Bbb R: \left\lVert (x,y) \right\rVert<r$}$ $ , to some r>0 and the norm is an euclidean norm, let $f(x,y)$ -> $L$ as (x,y) -> (0,0), with f : $B(o,r)$ -> $\Bbb R$. ...
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2answers
54 views

Proof of $\vec{\nabla}\cdot(\vec{\nabla}^2\vec{F}) = \vec{\nabla}^2(\vec{\nabla}\cdot\vec{F})$

How can I prove the following? $$\vec{\nabla}\cdot(\vec{\nabla}^2\vec{F}) = \vec{\nabla}^2(\vec{\nabla}\cdot\vec{F})$$ $$\vec{F}:\Bbb{R^3}\mapsto\Bbb{R^3}$$ I am confused because on the left part I ...
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1answer
34 views

Estimating mass of hurricane

Estimate the total mass of water (in kg) in a model hurricane using the following assumptions, being mindful of units. (a) The mass density of water $(g/m^3)$ is modeled as $$ p(r,z) ...
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1answer
35 views

$x^2+4xy-2xz-5y^2+6yz-3z^2\leq 0$ $\forall (x,y,z)\in\mathbb{R}^3$

I was thinking to check that the maximum is attached at a negative value, the function $f(x,y,z)=x^2+4xy-2xz-5y^2+6yz-3z^2$ is concave and so the local maximum is a global maximum, but how do I check ...
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1answer
55 views

Using higher order derivatives

I am currently learning about the general Notion of Differentiability. I came across some difficulties when working with higher order derivatives and I am hoping for confirmation or comments on some ...
1
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2answers
47 views

Functions of Several Variables - Successive Differentiation

I am struggling with this question: Regarding u and v as functions of x and y and defined by the equations : $$ x=e^u \cos(v) $$ $$ y=e^u \sin(v) $$ show that : $$\frac{\partial^2z}{\partial x^...
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0answers
17 views

What operations can be performed on/with a differential operator vs partial differential operator?

What operations can be performed on or with a differential operator vs a partial differential operator? I ask because I see people "divide" or "multiply" with the differential operator but that same ...
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2answers
37 views

When solving double integral, how do I find the range over where I evaluate the integral

For example: (1)Use the change of variables $x=u^2-v^2, y = 2uv$ to evaluate the integral $\int\int_R y dA$, where $R$ is the region bounded by the $x$-axis and the parabolas $y^2 = 4-4x$ and $y^2 = ...
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1answer
43 views

What's the difference between these two bullets?

Let $f:\mathbb{R}^3\to\mathbb{R}$. Get the equation of the tangent place to the level set: $\{(x,y,z)\in\mathbb{R}^3|f(x,y,z)=k\}$ in the point $p$ Get the equation of the tangent hyperplane to the ...
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1answer
83 views

Find $\theta$ and $\phi$ that maximize $\mid -2ia\sin\theta - 2ib\sin\phi + 2c(1-\cos\theta) +2d(1-\cos\phi)\mid$

How can you find for what values of the $\theta$ and $\phi$ angles the following modulus will assume its greatest possible value? $$\mid -2ia\sin(\theta) - 2ib\sin(\phi) + 2c(1-\cos(\theta)) +2d(1-\...
0
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1answer
29 views

For an orientable surface S and a fixed vector v, prove that…

Prove that $$2\iint_S v\cdot n dS = \int_{\partial S}(v\times r) dS$$ where $r=(x,y,z)$ and $n$ is the unit normal vector for $S$.
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1answer
33 views

Derivative of function defined as Integral

I have to find all partial derivatives of: $$ f(x,y,z) = \int_{\cos x + \sin y}^{z} e^{tz} dt $$ I easily get confused with all this variables, but the idea is to use The Fundamental Theorem of ...
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1answer
16 views

Find the absolute extrema of the function over the region R

Find the absolute extrema of the function over the region R where $f(x,y)=x^2+xy, R=\{(x,y)|\;|x|\leq2,|y|\leq1\}$, Here find we going find $f_x=2x+y=0, f_y=x=0 \Rightarrow (x,y)=(0,0)$ is i am ...
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0answers
31 views

Techniques for finding functions with known values and derivatives at two points

I want to find a function $\gamma(t) = (x(t),y(t))$ such that for two values of $t$, we have $\gamma'(t)$ and $\gamma(t)$ have some value, and at no point does the curvature ever exceed $r$. What ...
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2answers
69 views

Evaluating $\int_0^1\int_0^1 e^{\max\{x^2,y^2\}\,}\mathrm dx\,\mathrm dy$

The integral again for convenience is $$ I=\int_0^1\int_0^1 e^{\max\{x^2,y^2\}}\,\mathrm dx\,\mathrm dy $$ My thoughts: Ignoring for a moment that the region is a rectangle, I hoped moving to polar ...
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2answers
36 views

Triple Integral in Spherical Coordinates

I'm trying to solve the triple integral of $3x^2 + 3y^2 + 3z^2$ in spherical coordinates, with rho from 1 to 2, theta from 0 to 2$\pi$, and $\phi$ from 0 to $\pi /4$. Here's how I'm solving it: First ...
2
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1answer
37 views

How evaluate this integral in cartesian coordinates?

I can evaluate this with polar coordinates, but is it possible in cartesian coordinates? $$ \int\limits_{z=-1}^{1} \int\limits_{y=-\sqrt{1-x^2}}^{y=\sqrt{1-x^2}} \int\limits_{x=-\sqrt{1-y^2}}^{x=\...
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0answers
15 views

maximum of multi-variable function

I want to show that the following function attains maximum at $(1/2, 1/2, .....1/2)$ $$H(X) = \frac{1}{(1-\alpha)\beta}\sum_{i=1}^n[ ( x_i^{\alpha x_i} + (1-x_i)^{\alpha(1-x_i)})^{\beta} - 2^{\beta}]$$...
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0answers
82 views

Exact Probability of reducibility of Bivariate Polynomials

I am considering polynomials of the form $$P(x,y)= \sum_{k=0}^n\sum_{l=0}^n a_{k,l}x^{k}y^{l}$$ where $n \in \mathbb{N}$. The coefficients $a_{k,l}$ are considered to be randomly generated from the ...
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17 views

Is the unit normal really necessary to calculate the flux of a vector field?

So I've been studying Multivariable Calculus and this definition still eludes. Let $ \gamma:\mathbb [a,b] \rightarrow \mathbb R ^2$ be a closed, simple, regular curved. It is also oriented ...
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0answers
21 views

Quadratic taylor polynomial of $f(x, y) = x^4 + x^2y^2 + y^4$

Given $$f(x, y) = x^4 + x^2y^2 + y^4,$$ I would like to calculate the taylor polynomial of degree $2$ at $(0, 0).$ Approach This exercise confuses me. I simply tried to follow the ...
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2answers
101 views

surface area of $\left\{(x,y,z)\in R^3\,\mid\, x^2+y^2 =\frac{1}{z^2}, 1<z<3\right\}$

I want to calculate the surface area of the surface that bounds the solid $$K=\left\{(x,y,z)\in R^3\,\mid\, x^2+y^2 \leq\frac{1}{z^2}, 1<z<3\right\}$$ I'm stuck with the differential surface ...
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0answers
37 views

Justify why there can't be a strict local extremum at $\|x\|$ = $\sqrt{1 \over 2}$ for $f(x) = \|x\|^4 - \|x\|^2$

Given $f: \Bbb R^2 \rightarrow \Bbb R$ $$f(x) = \|x\|^4 - \|x\|^2$$ with $x := (x_1, x_2)$, I have to justify why there can't be a strict local extremum at $\|x\|$ = $\sqrt{1 \over ...
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2answers
58 views

Advanced Calculus Book for Computer Science Student

I study Computer Science, but our mathematic coures are a little bit to basic. I'm looking for a "advanced Calculus" book for self-study that has a lot of exercises. The book should focus on ...
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1answer
35 views

Triple integral question- two balls in $\mathbb{R}^3$

How can I calculate the volume bounded between the two surfaces: $$ x^2+y^2+z^2=1, \quad x^2+(y-1)^2+z^2=1 $$ and contains the point $(0,0.5,0)$. When I move to spherical coordinates, I obtain ...
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2answers
34 views

tangent to a level surface

Let $F:\mathbb{R}\to \mathbb{R}^n$ be differentiable. Let $f:\mathbb{R}^n \to \mathbb{R}$ be continuously differentiable and such that the composition $g(t)=f(F(t))$ exists. If $F'(t_0)$ is tangent ...
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1answer
54 views

Prove that $\lim_{\epsilon \rightarrow 0}\int_{\partial B_\epsilon} (φ∇g · n − g∇φ · n) ds = 2πφ(0, 0)$

Suppose $φ : \mathbb{R^2}\rightarrow \mathbb{R}$ is any $C^1$ function and let $g:\mathbb{R^2}-\{(0,0)\}\rightarrow \mathbb{R}$ given by $g(x, y) := \ln\sqrt{x^2+y^2}$ Prove that $\lim_{\...
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1answer
32 views

Have I made a mistake in the resolution?(surface integral)

Problem: Compute the surface area of that portion of the sphere $x² + y² + z² = a²$ lying within the cylinder $x² + y² = ay$ , where a > 0. My Try: We consider $r(x,y)=(x,y,f(x,y))=(x,y,\sqrt{a²-x²...
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2answers
28 views

Find the limit of the vector function

$lim_{t\to\infty} \Big(te^{-t},\frac{t^3+t}{2t^3-1},tsin(\frac{1}{t})\Big)$ a) $lim_{t\to\infty} te^{-t} = \infty \times 0$ $lim_{t\to\infty} 1e^{-t}+-e^tt = 0+(0\times\infty)$=undefined, and ...
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1answer
21 views

Find the volume of solid that lies under $z=x^2+y^2$, the $xy$ plane and inside $x^2+y^2=2x$

Find the volume of solid that lies under $z=x^2+y^2$, the $xy$ plane and inside $x^2+y^2=2x$ My attempt: $z = x^2+y^2 = r^2= 2x$, and $x=rcos(\theta)$, so $r^2=2rcos(\theta),$ $r = 2cos(\theta)$ $...
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1answer
23 views

Curve of intersection, value for parameter

This is for a line integral. Parametrize the curve of intersection: \begin{align*} S_1: x^2+4y^2 + z^2 &= 4a^2, y<0\\ S_2: x+2y &= 0 \end{align*} Orientation from $(0,0,-2a)$ to $(0,0,2a)$....
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2answers
63 views

At any $θ$ on curve $x=a\cos{θ}+aθ \sin{θ}$,$ y=a\sin{θ}-aθ\cos{θ}$ what is distance from the origin to nomal?

The parametric equation at any point $\theta$ of a curve is: $$x=a\cos{(\theta)}+a\theta\sin{(\theta)}, y=a\sin(\theta)-a\theta\cos{(\theta)}$$ What is its distance from the origin to its normal at ...
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1answer
40 views

How is the following derivative equal to the limit on the right side of the equation?

I am puzzled on how the following derivative is equal to the limit on the right side of the equation. I have tried to use the limit definition of a derivative to explain it, but I believe I am making ...
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1answer
71 views

Why does the gradient of matrix product $AB$ w.r.t. $A$ equal $B^T$?

The below passage is from p. 215 of Deep Learning by Goodfellow, Bengio and Courville. For example, we might use a matrix multiplication operation to create a variable $C = AB$. Suppose that the ...
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1answer
30 views

Find the volume common to $r^2+z^2=a^2$ and $r=a\sin(\theta)$.

Sorry to ask another one of these, but I am really struggling with these integrals. The question asks to find the volume common to $r^2+z^2=a^2$ and $r=a\sin(\theta)$. I attempted to set up the ...
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1answer
18 views

Hint needed in volume integral computation: Finding the volume in the first octant inside of $y^2+z^2=16$ but outside $y^2=3x$

I am looking for a hint about how to set up the bounds incorporating the information "outside of $y^2=3x$. I know that this should look like a cylinder along the $x$ axis, the parabola $y^2=3x$ ...
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1answer
40 views

Bounds of third integral

For integrals: $\iiint_D x^{2}yz \, \mathrm dx\mathrm dy\mathrm dz$, where $D$ is limited by surfaces: $x = 2, y=x^2, z=0, x+y=z$. $\iiint_D xz \, \mathrm dx\mathrm dy\mathrm dz$, where $D$ is ...
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2answers
31 views

evaluation of double integral using change of order of integration

How to evaluate the following double integral $\int\limits_s^t\int\limits_s^u e^{-\lambda(t-v)}(u-v)^{-\beta-1}dvdu $ where $\lambda$ and $\beta$ are positve constants.
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0answers
31 views

calculate area using stokes

Let $S:= \{(x,y,z) | z = -x-y \} \cap \{(x,y,z) | z^2+y^2+x^2 \leq 1\}$ I have to calculate the Area of $S$. Since $S$ is the unit circle. I know that the area is $\pi$ But I have to use Stokes. I ...
3
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0answers
48 views

Integral of determinant of Jacobian of a $C^{1}$ application.

Let $f \in C^{1}(\mathbb{R}^{n}; \mathbb{R}^{n})$, suppose that exists $r>0$ such that $f(x)=0$ if $|x|>r$. Prove that exists $k>0$ such that $\int_{\overline{B_{k}(0)}} det Jf(x) dx=0$. My ...
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1answer
106 views

An application $f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ and $C^{1}$ such that $f(x)=0$ for $x>r$ implies the value of jacobian integral is zero

Let $f \colon \mathbb{R}^n \to \mathbb{R}^n$ of class $C^{1}$. Suppose that exists $r>0$ such that $f(x)=0$ if $|x|\geq r$ .Prove that exists $k>0$ such that: $\displaystyle \int_{B[0,k]}$ det$...
0
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1answer
38 views

Difficulty setting up an iterated integral

I am trying to integrate the function $\frac{1}{\sqrt{2y-y^2}}$ over the region in the first quadrant bounded by $x^2=4-2y$. Given that this region is between bounded by an convex parabola and in the ...
0
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1answer
15 views

Question about terminology of Munkres's Analysis on Manifold text

The definition above are given by Munkres when he defined Euclidean manifold. One question I have about terminology is that when he says " For each $p \in M$, there is an open set $V$ of $M$ ..... ", ...
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1answer
26 views

2nd derivative of xy w/ respect to x?

$$\frac{d^2}{dx^2}xy$$ I know it equals zero but I don't know the in between-steps. I'm using it to prove Newtons Laws work in any frame of reference. So say two guys start from the same point and ...