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
22 views

finding largest value of x obtained from curve by implicit differentiation

Consider the curve defined by $x^2 + 2y^2 + 4 \beta x y = K $ with $K > 0$ and where $\beta$ is a (sufficiently small) parameter. Assuming that the above can be used to define a function $x = ...
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0answers
27 views

How does this optimization problem satisfy Karush-Kuhn-Tucker Conditions?

I am following Andrew Ng's course notes on Support Vector Machines at: http://cs229.stanford.edu/notes/cs229-notes3.pdf There is something in these notes which I do not understand. SVM's basic ...
0
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1answer
15 views

deriving formula for $y'$ in terms of various partial derivatives

Consider two three-variable functions $H(x, y, z)$ and $K(x, y, z)$ and the associated level surfaces $H(x, y, z)= a$ and $K(x, y, z)= b.$ It is assumed that these surfaces intersect along some curve ...
1
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1answer
53 views

Operator $\nabla$

My teacher told me to solve some physics problem where I need to find $\rho(r)$ by using: $$\frac{\rho(r).dV}{\varepsilon_0}=\vec E(r+dr)S(r+dr)-\vec E(r)S(r)$$ where $dV$ is a small hollow spherical ...
0
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2answers
22 views

difficulty proving this theorem

Let $x = s + t$ and $y= s - t$. For any $f(x, y)$, let us define this function in terms of s and t in the usual fashion: $g(s, t): = f(x(s, t), y(s, t))$. Show that $$ \left(\frac {\partial f} ...
0
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2answers
37 views

Simple question about finding the inverse function from $R^{1}$ into $R^{2}$

I'm stuck on computing the inverse for some simple functions; e.g., Consider the following two functions, $f(t) := [\cos(t), \; \sin(t)]$ with $t \in (-\pi, \pi)$ $g(x) := [x, \; x^2]$ ...
4
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2answers
49 views

Convolution integral problem

In the process of solving a certain PDE, I've arrived at a convolution integral: $$\int_{\mathbb{R}^3} G(x-y) \nabla p(y) dy$$ where $x \in \mathbb{R}^3$, $G(z)=\frac{1}{\| z \|}$ and $p(z) = ...
6
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1answer
32 views

Exists open subset and one-to-one $C^1$ mapping such that mapping of intersection is open subset

Let $M$ be a smooth $k$-manifold in $\mathbb{R}^n$. Given ${\bf p} \in M$, how would I go about showing there exists an open subsets $W$ of $\mathbb{R}^n$ with ${\bf p} \in W$, and a one-to-one $C^1$ ...
8
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1answer
23 views

Sum of $C_1$ mappings is one-to-one in neighborhood of a point

Let $f: \mathbb{R}^n \to \mathbb{R}^n$ be a $C^1$ mapping such that $df_{\bf a}: \mathbb{R}^n \to \mathbb{R}^n$ is one-to-one, so that $f$ is one-to-one in a neighborhood of ${\bf a}$. How would I go ...
0
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1answer
32 views

Show that $d\beta=0 \iff p=n/2$

Let $\beta$ be the $(n-1)$-form on $\mathbb{R}^n \setminus \{0\}$ given by $\displaystyle \beta = \sum_{i=1}^{n}(-1)^{i-1}\frac{x^i dx^1 \wedge dx^2 \wedge \dots \wedge \hat{dx^i} \wedge \dots ...
1
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2answers
25 views

How to differentiate this position Vector $\vec r=\rho\vec e_\rho+z\vec e_z$

Given the unit vectors: $\vec e_\rho=\bigl(\begin{smallmatrix} cos(\theta )\\ sin(\theta )\\0 \end{smallmatrix}\bigr); \vec e_\phi=\bigl(\begin{smallmatrix} -sin(\theta )\\ cos(\theta )\\0 ...
1
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0answers
30 views

Maximum / Minimum Cost of a Box

this is a sample final question for a multivariable calculus course. "A rectangular box has two opposing sides (left and right) made of gold, two (front and back) of silver, and two (top and bottom) ...
0
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1answer
18 views

Find the Moment of Inertia (Need Help With a Single Step)

Find the moment of inertia around the z axis of the region bounded by x=0, y=0, z=0, x+y+z = 1. My attempt at a solution: I believe that this should require a triple integral of the form ...
2
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1answer
31 views

Is a linear operator equal to it's jacobian?

In the course of proving something else, I was stuck and someone showed me that if the matrix which describes a linear operator $f: \Bbb R^n \to \Bbb R^n$ (and maybe $\Bbb C^n \to \Bbb C^n$?) is the ...
0
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0answers
14 views

Triple Integral with spherical coordinates

Find the volume of the region bounded by $(x^2 + y^2 + z^2)^2 = x$. I used spherical coordinates to get $p^3 = \sin \phi \cos \theta$ so does this imply that $ 0<= p <= (\sin \phi \cos ...
0
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1answer
19 views

How to compute the derivative of composition $F(t) = f(A(4t, t), A(t, 2t))$?

Let $f(x, y)$ denote a function of two variables whose partial derivatives – $f_x(x, y)$ and $f_y(x, y)$ – are considered to be “known” functions. Let $A(x, y)$ be another function whose partial ...
0
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1answer
40 views

Piecewise defined function of two variables that has partial derivatives but is not differentiable

Let $$f(x,y)=\cases{0 & if $x=y$\\7x-3y & otherwise }.$$ Show $f_1(0,0) = 7$ and $f_2(0,0) = -3$, but $f$ is not differentiable at $(0,0)$. I found $f_1$ and $f_2$ by definition of ...
0
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0answers
15 views

Integrals over a Surface Using Stokes Theorem, C not parallel to coordinate axes

F(x, y, z) is a vector field, specifically, F (x, y, z) = 2zi + xyj + 4yk C is the ellipse obtained by intersecting the plane y + z = 4 with the cylinder x^2 + y^2 =4 oriented in a counterclockwise ...
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0answers
24 views

Integrals over a Surface Using Stokes Theorem

Let $$\vec{F} (x, y, z) = xy\hat{i} +(4x - yz)\hat{j} + (xy - z^{1/2}) \hat{k},$$ and let $C$ be a circle of radius $r$ lying in the plane $x + y + z = 5$. If $$\int_C \vec{F} \cdot d\vec{r} = \pi ...
0
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1answer
13 views

Find an Equation of the Tangent Plane to the Given Parametric Surface at the Specified Point.

I am given $x=u+v$, $y=3u^2$, and $z=u-v$. I need to find the equation of the tangent plane at $(2,3,0)$. I understand that the equation of the tangent plane is ...
1
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1answer
36 views

Line integral of second kind over a circle: $\int \frac{xdy - ydx}{x^2+y^2}$

I've just get stuck with some task of line integral: $$\int \frac{xdy - ydx}{x^2+y^2}\quad \text{ over} \ x^2+y^2=R^2$$ I understand that I need to use polar coordinates, and I have such thing: $$x ...
0
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2answers
75 views

Finding the volume bounded by surface $y^2=4ax$ and the planes $x+z=a$ and $z=0$

The problem is stated below: Let $V$ be volume bounded by surface $y^2=4ax$ and the planes $x+z=a$ and $z=0$. Express $V$ as a multiple integral, whose limits should be clearly stated. Hence ...
2
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0answers
20 views

multivariable calculus charge density question

The sphere given by $x^{2} + y^{2} + z^{2} = 4$ is submerged in an electric field with charge density given by $f(x, y, z) = x^{2} + y^{2}$. Find the total amount of electric charge on this surface. ...
1
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1answer
32 views

What's the fourth term in the multivariable Taylor expansion?

For a function $f: \Bbb R^n \to R$, the $2$nd order Taylor expansion is: $$f(\mathbf x+\mathbf h) \approx f(\mathbf x)+ Df(\mathbf x) \mathbf h + \frac{1}{2}\mathbf h^T H(f)(\mathbf x) \mathbf h$$ ...
0
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1answer
16 views

Using $q_A(x)=x^tAx$, the quadratic form associated with $A$.

In my notes on finding the local extrema ,I had the following extract : Let $A$ be an $n\times n$ matrix ,a principal submatrix of size $k$ is the submatrix with first $k$ columns and first $k$ ...
1
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1answer
46 views

proving gradient of a function is always perpendicular to the contour lines

Can someone give an explanation of how such a proof would go, given a function example: $y = f(x)$
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2answers
31 views

if $f(x,y)=\phi(x-cy)+\phi(x+cy)$ then $f_{22}=c^2f_{11}.$

Any hint how to solve this: Show that if $f(x,y)=\phi(x-cy)+\phi(x+cy)$ then $f_{22}=c^2f_{11}.$ I don't know how to proceed with this .Kindly help with some hint how to solve this...
1
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1answer
31 views

Vector Fields problem unsure of how to start

A fluid having density $f(x, y, z) = x^{2} + 2y^{2} + z^{2}$ flows with velocity $v(x, y, z) = x^{2}i + xy^{2}j + zk$. Determine the rate of mass flow through the sphere $ρ = 1$ in the outward ...
3
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1answer
25 views

definition of differential does not depend on the choice of the curve, differential is a linear map

Let $\varphi: M \to N$ be a differentiable map. How do I show that the definition of the differential $d\varphi_p: T_pM \to T_{\varphi(p)}N$ of $\varphi$ at $p$ does not depend on the choice of the ...
2
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0answers
20 views

Computing $F_{12}$ when $F(x,y)=f(x,xy);F(x,y)=f(x,y,g(x,y))$

I've a question as follows: Compute $F_{12}$ when $F(x,y)=f(x,xy);F(x,y)=f(x,y,g(x,y))$ I think it to be an application of implicit function theorem . but don't know how to apply, kindly ...
1
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0answers
38 views

When the function f is constant?

Let $ f: \mathbb R^n \to \mathbb R^n$ be a differential function. Let $Df(x)$ be the derivative of $f$ at $x$ in $\mathbb R^n$. Then which of the following is/ or correct? $Df(0)(u) = 0$ for all $u$ ...
3
votes
1answer
30 views

Every immersion is locally an embedding?

Let $\varphi: M \to N$ be an immersion and let $p$ be a point in $M$. How would I go about showing that there exists a neighborhood $V \subset M$ of $p$ such that the restriction $\varphi|V$ of ...
0
votes
1answer
24 views

Limit of Implicitly Defined Function (Follow up)

Is there a method such that one can determine the limit $\lim_{h \to 0}\frac{f(x)}{x^n}, n \in \mathbb{Z}^{+}$ for an implicitly defined function, defined near the origin - such as ...
1
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1answer
26 views

Multivariate function maximum criterion

Be a concave mutivariate function $f(\textbf{x})=\textbf{y}$. I observed the following conjecture: the maximum value of $f$ is achievable when all entries of $\textbf{x}$ are equal. How to prove such ...
4
votes
1answer
55 views

Doubt in solution for evaluating $\int_0^1\int_0^1\int_0^1(1+u^2+v^2+w^2)^{-2}du~dv~dw$.

I've a doubt in the answer for evaluating the following integral: $$\int_0^1\int_0^1\int_0^1(1+u^2+v^2+w^2)^{-2}du~dv~dw$$ solution: call this integral as $I$ .By symmetry we may compute it ...
4
votes
1answer
22 views

no point where assumptions of Inverse Function Theorem hold?

Let $\varphi: \mathbb{R}^3 \to \mathbb{R}$, $\psi: \mathbb{R}^3 \to \mathbb{R}$ are continuously differentiable. Define $f: \mathbb{R}^3 \to \mathbb{R}^3$ by$$f({\bf x}) = (\varphi({\bf x}), \psi({\bf ...
2
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1answer
26 views

Doubt on integrating $f$ on a given region .

I was asked to integrate the following function: $f(x,y,z)=1-z^2$ on $U$ where $U$ is a pyramid with the top vertex $(0,0,1)$ and base vertices :$(0,0,0),(1,0,0),(0,1,0), (1,1,0)$ the ...
1
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0answers
31 views

Sine wave from fos + simple signal

I have a first order system $\frac {1}{(s+c)}$ and a signal of the form $\sum_{k=0}^\infty (-1)^{k}e^{-2ks}a(\frac{1-e^{-2s}}{s}- be^{-s}(\frac{1-e^{-s}}{s^2})) $ i.e a periodic signal of a square ...
2
votes
1answer
22 views

Find $\iint\limits_\Sigma(x\sqrt{x^2+y^2+z^2}\hat{\imath}+y\sqrt{x^2+y^2+z^2}\hat{\jmath}+z\sqrt{x^2+y^2+z^2}\hat{k})\cdot\hat{n}dS$

Find $\iint\limits_\Sigma(x\sqrt{x^2+y^2+z^2}\hat{\imath}+y\sqrt{x^2+y^2+z^2}\hat{\jmath}+z\sqrt{x^2+y^2+z^2}\hat{k})\cdot\hat{n}dS$ Where $\Sigma$ is the torus $\rho=\sin\phi$ Oriented by the ...
2
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0answers
27 views

If $f$ is integrable on $S_1$ and $S_2$ ,then $f$ is integrable on $S_1$\ $S_2$ . [closed]

Can anyone give some hint how to proceed with the problem below: Let $S_1$ and $S_2$ be bounded simple sets in $\mathbb R^n$ .Let $f:S_1\cup S_2:\to\mathbb R$ be bounded function.Show that if ...
0
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0answers
13 views

Problem-solving method for calculating flux on a half-cylinder

If I wanted to calculate the flux on a half-cylinder's surfaces: $$\iint_S\mathbf F\cdot d\mathbf s$$ The easiest way to go about this is to use divergence theorem, I found, so that I get ...
6
votes
1answer
119 views

Differential forms, number of zeros on disk

Let $f$ be a holomorphic function on $U \subset \mathbb{C}$ and $f'(z) \neq 0$, $z \in U$. Then how do I show that all zeros of $f$ are simple and positive, and that if I have a disk $D \subset U$ ...
0
votes
1answer
40 views

Prove that this limit equals to 2.

Prove $$\lim_{(x,y)\to(0,0)}\frac{\sin(x^2+y^2)}{1-\cos(\sqrt {x^2+y^2})}=2$$
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0answers
13 views

Show that Stokes's theorem holds for the vector field $F=6xy\mathbf{i} + (z+1)^2\mathbf{j} + 2y\mathbf{k}$

Show that Stokes's theorem holds for the vector field $F=6xy\mathbf{i} + (z+1)^2\mathbf{j} + 2y\mathbf{k}$ and the surface S lying in the plane $z=0$ bounded by $1\leqslant x^2+y^2\leqslant4.$ ...
1
vote
2answers
40 views

Show that this two variable limit don't exist

Prove that the following limit doesn't exist: $$\lim_{(x,y)\to(1,0)}\frac{\sin(1-x^2+y^2)-y}{xy}$$ I guess I need to find different values for different directions.
1
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0answers
16 views

Surface area of this helicoid?

What is the surface area of the paramatized helicoid $$X(r,s)=(r\cos(s), r\sin(s), s)$$ with $0\leq r \leq 1$ and $0\leq s \leq 2n\pi$, where $n$ is a positive integer? I attempted to take the ...
0
votes
0answers
19 views

How to compute $\int_C (X|dx)$?

I have the vector field $X(x,y,z) = (\frac {-y}{x^2+y^2+z^2}, \frac{x}{x^2+y^2+z^2},0)$ on $\mathbb{R} ^3$ \ $\left\{ (0,0,0) \right\}$. I need to compute $\int_C (X|dx)$ over a closed circle of ...
1
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1answer
26 views

Finding the integral using divergence theorem without vector field

Use the divergence theorem to evaluate $$\iint_S (x^2 y^2+y^2 z^2+x^2 z^2)\ dS,$$ where $S$ is the entire surface of the sphere of unit radius, centered at the origin. How do I find this? ...
0
votes
0answers
6 views

Trouble Evaluating Double Integral from Patramterized Surface

The function is $\frac{y^3}{3}+\sqrt{2}x$, and $x^{1/3}$ to 1 is the bounds for y, and x goes from 0 to 1. So I know that the Surface Area of a surface is found by $\iint{||T_uxT_v||}dudv$, so I ...
1
vote
1answer
23 views

Comparison of the change of variable theorem

I would like to compare the change of variable theorem for 1 variable and more. What are the differences, in which case we need stronger assumptions? How do they differ? What is the best way to write ...