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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24 views

Clarification on to a Solution using Stokes Theorem

The question is there as well, the only part missing is that $F=r$x$(i+j+k)$, where $r=(x,y,z)$. So then $F=(y-z, z-x, x-y)$. The rest is there. I did and understand the problem until after the ...
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1answer
22 views

Double Integrals and Volume [on hold]

How do I find the volume of a solid in the first octant bounded by the coordinate planes and the graphs of the plane z=3-x-y and the cylinder $x^2+y^2=1$?
3
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1answer
89 views

A Deviation from a Conventional Proof of the Basel Problem

There's been many topics on the Riemann-Zeta function, specifically $\zeta(2)$.$$\zeta(2)=\sum_{n=1}^\infty\frac{1}{n^2}=\int_0^1\int_0^1\frac{1}{1-xy}dA$$This is the Basel Problem. Taking the ...
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0answers
21 views

How to represent real algebraic numbers with period integrals

Background: A real period is defined to be the value of an integral of the form $$\int_D R(x_1,\cdots,x_n)dV$$ where $R$ is a rational function with rational ceofficients, and $D\subseteq\Bbb R^n$ ...
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0answers
11 views

Expressing integrals of rational functions over domains defined by rational inequalities as differences of volume integrals

Background: A real period is defined to be the value of an integral of the form $$\int_D R(x_1,\cdots,x_n)dV$$ where $R$ is a rational function with rational ceofficients, and $D\subseteq\Bbb R^n$ ...
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0answers
12 views

Question from Stewart's Calculus regarding proof of independence of path and conservative vector fields.

Please look over this proof. In the proof, it says: "Notice that the first of these integrals does not depend on $x$, so..." How is that so? $C_1$ does depend on $x$. How/Why does it not ...
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1answer
33 views

Taylor's theorem in two variables? [on hold]

What would be the expression for the second order Taylor’s formula near a critical point? will it be same as in case of any general point or somewhat different in case we talk about critical points? ...
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1answer
32 views

Finding the volume of a solid under a region (Triple integrals)

Let S be the solid enclosed above by $x^2+y^2+z^2=2$, below by $x^2+y^2=z^2$ and $y=0$ compute the integral $$ \iiint_S \frac{z}{\sqrt{x^2 +y^2 +z^2} }\text{d}x\,\text{d}y\,\text{d}z $$ What i tried ...
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0answers
16 views

Finding directional derivatives at $(0,0)$ Multivariate calculus

Given the following function $$f(x,y)=\begin{cases}\frac{x^{707}y}{x^{1414}+y^2}, &(x,y) \neq (0,0)\\0 ,&(x,y)=(0,0) \end{cases}$$ Does $f$ have a directional derivative in every direction ...
1
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1answer
28 views

Finding critical points of a multivariable function

Let $f(x,y)=e^{x^2-xy+y^2}$ (a) Find all the critical points of the following function. (b) Find the all the local maxima and local minima of the function if there is any. What i tried. I tried ...
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2answers
15 views

Showing that normal line passes through a point.

I need to show that a line passes through a point. How should I go about doing this? The question is: Let $L$ be the normal line at $(1,1,1)$ to the level surface of $f(x,y,z) = x^2 - z$ that ...
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1answer
17 views

Finding Absolute Min/Max with given Domain and Equation. f(x,y)

Question is: Suppose that $f(x,y) = 5x+3y$ at which $-3 \leq x$, $y \leq 3$. Find Absolute minimum and maximum of $f(x,y)$. Since $\frac{\partial}{\partial x} f(x,y) = 0$ or ...
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0answers
6 views

Needs clarification about statements related to continuity , differentiability (Multivariable)

I have recently done limits,continuity of multi variable functions ,but i feel i need clarification as to which is true or not regarding these statements . Which of these are true ? A . If partial ...
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0answers
33 views

Solving the PDE $\frac{\partial u}{\partial t}=a\frac{\partial^2 u}{\partial x^2}+b\frac{\partial u}{\partial x}$

I am trying to solve the PDE $\frac{\partial u}{\partial t}=a\frac{\partial^2 u}{\partial x^2}+b\frac{\partial u}{\partial x}$ for constants $a$ and $b$ with conditions $\frac{\partial u}{\partial ...
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1answer
13 views

Finding derivative at specific time on space curve

I am trying to do some practice questions in my book, but I don't know how to do this specific question: Suppose the function $F(x,y,z,t)$ satisfies $F_x(3,9,18,3)=1$, $F_y(3,9,18,3)=-2$, ...
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1answer
44 views

One Question about the Fubini's Theorem

The Fubini's Theorem says: If function $f:X \times Y \rightarrow R$ is integrable over $X \times Y$, then $$ \int_{X \times Y}f(x,y)dxdy = \int_{X}dx\int_{Y}f(x,y)dy = \int_{Y}dy\int_{X}f(x,y)dx. $$ ...
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0answers
32 views

Fun Lagrange multiplier problem?

Do any of you have a fun or interesting Lagrange multiplier problem that would be suitable for undergraduate calculus students? I'm planning on working through a standard Lagrange multiplier problem ...
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0answers
16 views

Does integrating a gradient give the original function?

I just want to make sure that this is true before using it in a problem, thanks
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1answer
31 views

Continuity of a function at $0$

A similar has been asked before, but it was confusing. Please help me with it. I need a general method of dealing with such problems I need to show that the following function is continuous at $0$. ...
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0answers
19 views

Implicit partial second derivatives from coupled equations

I have three functions defined in two variables. $f(x,y)$ $g(x,y)$ $h(x,y)$ I wish to find the partial derivatives $f_{gg}$, $f_{hh}$, and $f_{gh}$ and evaluate them at a particular point. In this ...
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0answers
20 views

Non-linear integral equation

Show that the function $$x(t) = \frac{1}{{\sqrt {k \cdot m} }} \cdot \int_0^t {F(\tau ) \cdot \sin \left( {\sqrt {\frac{k}{m}} \cdot (t - \tau )} \right)\,d\tau } $$ satisfies the initial conditions ...
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3answers
20 views

Triple Integrals and Spherical Coordinate Grid

I understand triple integrals in cartesian and 2D/3D cylindrical polar coordinates (at least I think I do) because I can visualize their coordinate grids. Take the usual $x-y$ coordinate grid, stack ...
1
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1answer
17 views

Showing that a vector field $\vec{G} = (F_2(y, x), F_1 (y, x)$ is conservative given that $\vec{F} = (F_1 (x, y), F_2 (x, y))$ is conservative

The question is in the title. I have tried a huge amount of counter-examples and have come to the conclusion that the vector field is conservative. It can be shown by counter-example that the vector ...
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3answers
55 views

Intuitive understanding of path integral formula

I have learned a general formula for a path/line integral $$ \int_a^b f\left(\mathbf{r}(t)\right) \|\mathbf{r}'(t)\|\ dt \tag{1} $$ and I'm trying to better understand it. Specifically, I'm ...
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2answers
13 views

First derivative of a multivariable function

This question was featured on my calculus mid-term today : Find the first derivative of $g(x,y)$ where : $$ g(x,y) = f(x^2 + y^2 ,xy) $$ This is the exact text of the problem. I just don't ...
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1answer
25 views

Convert to polar and evaluate

I have $$z= x^2 + y^2$$ $$z=2x$$ I set them equal to get their intersection, I get $$2x= x^2 + y^2$$ $$0= x^2 -2x +y^2$$ by completing square I get $$y= \pm \sqrt{1-(x-1)^2}$$ I need to put ...
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2answers
33 views

Generalized forms of Curl and Divergence

The definitions I learned in my calculus courses for curl and divergence were rather, at least to me, unintuitive and seemed to work only for $\mathbb{R}^3$. I took a look on Wikipedia: "The ...
3
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1answer
23 views

Setting up the intergal but do not integrate

I'm having a little trouble with this problem. Let D be the solid bounded by y=x, z=1-y^2, x=0, and z=0 1) Sketch the region of integration using 2 and dimensional sketches to show the region ...
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1answer
30 views

Mutlivariable Calculus: Surface Area

This was a question a students had asked me earlier today regarding surface area. Find the surface area of the hemisphere $x^2+y^2+z^2 = 4$ bounded below by $z=1$. I decided to approach ...
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1answer
33 views

Find the center of mass of soda can?

Help finding center of mass of soda can? If you represent the soda can as a right-circular cylinder radius=4 cm height =12 cm We are told to neglect the mass of the can itself. When the can is ...
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0answers
25 views

Surface area with double integral - how to parameterize?

Problem: Find the surface area of the part of the cylinder $x^2+z^2 = a^2$ that is inside the cylinder $x^2+y^2 = 2ay\;$ and also in the positive octant $( x \ge 0, y\ge 0, z\ge 0$). Assume $a > ...
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1answer
39 views

Find average z coordinate..Help please?

Question: Find the average z coordinate of all the points on AND within a hemisphere of radius 4 centered at the origin, and with it's base in the xy-plane. So I am assuming the function will be ...
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2answers
35 views

Differentiation problem, hard! [closed]

Having some trouble getting my head around the method of this question. Please help. find $dw/du$ and $dw/dv$ using the chain rule. if: $w = 4/(x^2 +y^2 +z^2)$ $x = u^2 + v^2$ $y = u^2 + v^2$ ...
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28 views

Help needed on Green's theorem.

Using Green's Theorem, find the area of the region formed by the intersection of $$x^2 + y^2 \leq2 \text{ and } y\geq1$$.
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1answer
40 views

Stokes’ Theorem to find integration

Use Stokes’ Theorem to evaluate integration $c (xy \,dx+ yz\, dy + zx\, dz)$ where and $C$ is the triangle with vertices $(1,0,0),(0,1,0),(0,0,1)$, oriented counter-clockwise rotation as viewed from ...
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0answers
15 views

Finding the value of the inverse function with inverse function theorem

I am stuck by the following problem. Let $h:\Bbb R^2\rightarrow \Bbb R^2$ and $$h(x,y)= (x^2+3xy+xy^3, x^3-5y^2)$$ Let $g=h^{-1}$ near $(0,1)$. Find $Dg(0,-5)$ I showed that the inverse function ...
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1answer
39 views

obtaining a surface equation by rotation

Let $C$ be a curve on the plane $(xoz)$ given by equation $g(x,z)=0$. How to find the equation in cylindric coordinates of the surface obtain rotating $C$ around zz axis?
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1answer
26 views

Can you switch the order of the determinants when changing variables using the Jacobian?

Let say we're changing the variables and we use the Jacobian to do this. Lets say we integrate in respect to $u$ and $v$, does it matter if we set up the integral like ...
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0answers
26 views

proof: $f ( x,y,z ) = x^2 + y^ 2 + z ^2$ is continuous function [closed]

proof: $f ( x,y,z ) = x^2 + y^ 2 + z ^2$ is a continuous function as the title say, thanks
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0answers
58 views

How to find the partial derivative of a row vector?

If I have $\vec{x} = \begin{pmatrix}x_1\\ x_2\\ \vdots\\ x_n\end{pmatrix}$, how can I find the derivative of $\vec{x}^T$? This questions comes when I was trying to find the minimum of the inner ...
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1answer
26 views

What happens to other eigenvalue? - steady state heat equation

A circular plate is bounded by circles of radii $r=2$ and $r=4$, its surface is insulated and temperatures along boundaries are given by $u(2,\theta)=10\cos\theta + 6\sin\theta$ and ...
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1answer
48 views

Evaluate Double Integral

$$ \int _0^{\sqrt{\pi}} \int_y^{\sqrt{\pi}} \sin (y^2 )\; dydx$$ Even if I change the order of integration I don't see how to get rid of this $\sin (x^2)$ which doesn't have antiderivate. It is ...
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0answers
14 views

Converse of Euler's homogenous function theorem proof follow up question

So basically I read someone else's answer to a question regarding Euler Homogeneous function theorem source: http://quant.stackexchange.com/questions/8911/what-is-exactly-eulers-decomposition Also ...
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4answers
72 views

Find the limit $\lim_{(x,y,z)\to(0,0,0)}\frac{xy+xz+yz}{x^2+y^2+z^2}$

Find the limit if it exists $$\lim_{(x,y,z)\to(0,0,0)}\frac{xy+xz+yz}{x^2+y^2+z^2}$$
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1answer
41 views

$n$-th derivative of $\displaystyle \left(\sum_{i=1}^Nx_i\right)^n$

I have no idea how to find the $n$-th derivative of $f:\mathbb{R}^N\rightarrow\mathbb{R},x\mapsto $ $(\sum_{i=1}^N x_{i})^n$. I tried to use the multinomial theorem, as well as only the chain rule. I ...
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2answers
23 views

Studying the differentiability of a function at a point $(a_{1},a_{2})$

I have a function $\ f: \mathbb{R}^2 \to \mathbb{R} $ to study: 1) It's continuity at the point $(a_1,a_2)$. 2) The partial derivative exists at $(a_1,a_2)$? 3) Are the partial derivatives ...
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3answers
17 views

Total surface area of a right circular cone [closed]

I do not know how to derive the total surface area of a cone whose formula is πrl+πr². Please guide me.
3
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2answers
53 views

Show that constant curvature $\kappa = 1/r$ is necessary and sufficient that the curve is a circular arc of radius $r$

We have to prove that a curve has constant curvature $\kappa = 1/r$ if and only if it is in a circular arc of radius $r$. I am confused because doesn't a helix also have a constant curvature given ...
0
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1answer
23 views

Double Integral Confusion..

for the double integral $\iint$$5xye^{(-x^2)}dxdy$, are you able to pull out the $5y$ and then integrate over $xe^{(-x^2)}$? For the indefinite integral I got $5y(\left(\frac12\right)e^{(-x^2)})$, ...
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3answers
51 views

Evaluating the following integral: $ \iiint_R x \, dA$ using a change of variable.

I'm calculating the following integral: $$ \iiint_R x \, dA$$ Where $S$ is the region in the first quadrant delimited by the portion of the circle $x^2+y^2=4$ and the lines $y=1$ and $y=2$. This ...