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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2answers
192 views

Write the function in the form $y=f(u)$ and $u=g(x)$. Then find $dy/dx$ as a function of $x$

$$y=\left(3x^2-(8/x)-x\right)^9$$ I know that $y = u^9$ and then $u = 3x^2-\dfrac{8}{x}-x$, but then I do not know how to put it together to solve for $dy/dx$.
5
votes
0answers
106 views

When does Gâteaux imply Fréchet? [duplicate]

Speaking of the relation between Gâteaux and Fréchet, authors usually point out that $$\text{Fréchet} \implies \text{Gâteaux}$$ and then give a counterexample to illustrate that the converse doesn't ...
2
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4answers
133 views

What is $ \lim_{(x,y)\to(2,2)}\frac{x^4-y^4}{x^2 - y^2} $?

I have limit: $$ \lim_{(x,y)\to(2,2)}\frac{x^4-y^4}{x^2 - y^2} $$ Why is the result $8$ ?
0
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1answer
57 views

how to represent this domain of integration

I have an exercice from a Stewart's book, I don't have the book with me and I don't remember the number and the page... so the question is to evaluate : $$\int_{1/ \sqrt 2}^1 \int_{\sqrt{1-x^2}}^x xy ...
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1answer
70 views

parametrize curve rotating about a line

I'm thinking of parametrizing a surface of revolution created by rotating $y=x^3, 0<x<1$ about the line x = 1. My attempt is let $z=x^3$ and $|x-1|$ be the radius of circle generated by ...
1
vote
1answer
31 views

Write the line integral of a vector field F over a boundary C as a sum of three one-variable integrals with correct limits and integrands?

Let S be the paraboloid $z = 5x^2 + 3y^2$ in $\mathbb{R}^3$ lying over the region $R$ in the $xy$-plane bounded by the lines $x+y=3$ and the coordinate axes. Suppose that the orientation of $S$ is ...
0
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1answer
76 views

Conservative Vector Fields

One of the theorems for a vector field to be conservative is that $$\frac{\partial N}{\partial x}=\frac{\partial M}{\partial y}$$ for $$F=\langle M,N\rangle.$$ To find the $$\int ...
5
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1answer
94 views

Volume of neighborhood of the curve

Let $\gamma:[0,1] \to \mathbb R^n$ be a smooth closed curve, such that $\gamma(0)=\gamma(1), \gamma'(0)=\gamma'(1), |\gamma'(a)|=1$ and let $B_r=\{x| \exists t, |\gamma(t)-x|<r\}$. How can i show ...
0
votes
1answer
74 views

Find the volume of the region with triple integrals.

The volume of the region bounded by $y^2+z^2=1\ \text{and} \ z^2+x^2=1$ should be found. Cylindrical conditions are perhaps the most appropriate in this case but the limits of the integral are what ...
1
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1answer
110 views

Stokes' Theorem: line integrals around 2-faces of n-dimensional surface?

Suppose we have a convex polytope in $n$ dimensions and are trying to calculate the surface integral (over this polytope) of some scalar function $f:R^n \rightarrow R$. Suppose all edges and vertices ...
1
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1answer
51 views

Use spherical coordinates to evaluate

Use spherical coordinates to evaluate $\int_{-2} ^{2} \int_0 ^{\sqrt{4-y^2}} \int_{-\sqrt{4-x^2-y^2}} ^{\sqrt{4-x^2-y^2}} y^2\sqrt{x^2+y^2+z^2} dz \ dx \ dy$ I did like this. Is that right ? ...
1
vote
1answer
394 views

Integral transformation change of variables

Use the change of variables formula and an appropriate transformation to evaluate $\int \int_R xy \ dA$ , where R is the square with vertices $(0,0), (1,1),(2,0)$ and $(1,-1)$. The answer is 0. Can ...
4
votes
2answers
77 views

Question on Curl F

The problem in the book asks what the curl of $\operatorname{curl}\vec F(\vec r)= \frac {\vec r}{\|\vec r\|}$. Can someone give me a good explanation on why the curl will be zero? I would really ...
2
votes
1answer
61 views

Find multiple integrals $I_{\max}(k,n)$ and $I_{\min}(k,n)$ in various ways

$I_{\max}(k,n)=\underbrace{\int\limits_0^1\int\limits_0^1\dots\int\limits_0^1}_k\left(\max\limits_{1\le i\le k}x_i\right)^n\,dx_1dx_2\dots dx_k$ ...
0
votes
2answers
36 views

Doing a Line Integral Problem

Here is my attempt: $$W=\int_C\vec{F}\cdot d\vec{r}\\=\int_C\frac{\alpha x}{(x^2+y^2)^{3/2}}dx+\frac{\alpha y}{(x^2+y^2)^{3/2}}dy\\Using\quad x=2t+1\quad and \quad y=-2t\quad for\quad 0\le t\le ...
1
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2answers
81 views

combining Gauss and Stokes theorems leads to nonsense

Gauss Theorem: $$\int_S \vec{a} \, d\vec{S}=\int_V\operatorname{div}(\vec{a})\,dV$$ Stokes theorem: $$\int_C \vec{a}\,d\vec{l}=\int_S\operatorname{curl}(\vec{a})\,d\vec{S}$$ Combining together: ...
3
votes
2answers
362 views

Differentiability of $g:=f(\sqrt{x^2+y^2})$ for a $C^1$ function with $f'(0)=0$ NBHM $2008$

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a continuously differentiable function such that $f'(0)=0$. Define for all $x,y\in \mathbb{R}$, $$g(x,y)=f(\sqrt{x^2+y^2})$$ Pick out the true statements ...
0
votes
3answers
404 views

Changing from Cartesian coordinates to Polar coordinates

Rewrite the iterated integral $$\int_0^1 \int_0^{\sqrt{2y - y^2}} (1 - x^2 - y^2)\,dx\,dy$$ in polar coordinate form. Do not evaluate the integral. Here is my answer: ...
2
votes
4answers
133 views

How to integrate this double integral?

$$\iint \limits_D 2x^2e^{x^2+y^2}-2y^2e^{x^2+y^2} dydx $$ where D is the region $x^2+y^2=4$ I tried changing it to polar, but it didn't make any use. $\iint \limits_{D(r,\theta)}2r^3\cos2\theta ...
0
votes
1answer
38 views

Calculate the integral on a closed, smooth curve.

$$\oint \limits_C (7y+x+2)dx+(5+y+2x)dy$$ where the curve C is the circle: $(x-a)^2 +(y-b)^2=25$. This integral calls for Green's Theorem. $$\iint \limits_D-5dydx$$ I beleive the region D is best ...
0
votes
1answer
192 views

Finding the area of the region with double integrals

I have to find the area of the region inside $r^2=16\cos(2\theta)$ and inside $r=2\cos(\theta)$. Should I divide the positive x and y region into two parts? Or can I bound r by ...
0
votes
1answer
66 views

Extreme values of a multivariable function

I am studying multivariable calculus and i don't know how to find the extreme values on a specific restriction ,i.e, $f(x,y,z) = x^2 + y^2 - z$ on the restriction $2x - 3y + z - 6 = 0$ . please help ...
2
votes
5answers
256 views

Prove that $2\int_a^b \int_a^x f(x)f(y) \, dy \, dx = \left[ \int_a^b f(x) \, dx \right]^2$

Suppose $f$ is a continuous single-variable function, prove that: $$2\int_a^b \int_a^x f(x)f(y) \, dy \, dx = \left[ \int_a^b f(x) \, dx \right]^2$$ This question was just on my Calculus III final ...
0
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1answer
170 views

Calculus 3 Finding the minimum distance

Find the minimum distance between two parabolas $y=x^2+2$ and $x = y^2$. Hint: parameterize the first parabola as $r_1(t) = (t,t^2+2)$ and the second parabola as $r_2(s) = (s^2,s)$ Not a clue on how ...
0
votes
1answer
97 views

What direction does the n vector (normal to the surface) have to be when doing Stokes' theorem?

The author uses $g=y+z-2$ instead of $g=2-y-z$ to ensure that n has a positive k component so that it points outward. But why was it necessary that n points outward? Is it because C is in the ...
2
votes
7answers
3k views

Why does $r=cos\theta$ produce a circle?

I am trying to do a double integral over the following region in polar coordinates: I know that the limits of integration are: $$\theta=-\pi/2\quad to\quad \theta=\pi/2\\r=0\quad to\quad ...
0
votes
1answer
63 views

Changing the order of integration in triple integral

Will you please help me change the order of integration in the following? $\int_{0}^{1}dx \int_{0}^{1}dy \int_{0}^{x^2+y^2} f dz$ . we need: $\int_{?}^{?}dz \int_{?}^{?}dy \int_{?}^{?} f dx$ ...
0
votes
1answer
54 views

Improper Multivariable Integrals

How can I find the values of $\alpha$ for which the following integrals (in $\mathbb{R}^n $ ) converge ? $\int_{|\vec{x}|\geq 1 } \frac{ln(|\vec{x}|^3 )}{|\vec{x}|^\alpha} d\vec{x} $ ...
1
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1answer
40 views

Triple integral over spherical area

Question: $$\iiint_S\sqrt{x^2+y^2}dxdydz$$ $$S: x^2+y^2+z^2\le9$$ $$0\le z$$ I've solved it to $\frac{81\pi^2}{8}$ by using spherical coordinates, but I got in my head that I should be able to ...
0
votes
0answers
165 views

Changing the Order of Integration in this Triple Integral

Evaluate $ \iiint_D (x^2+y^2) \, ,\mathrm{d}V $, where $D$ is the region bounded by the graphs of $y=x^2$, $z=4-y$, and $z=0$. So after over at least an hour of thinking, I might have all 6 ...
0
votes
2answers
73 views

Deriving an equation for solid of revolution

I was wondering, if there is any generic method that will help me find an explicit formula for a region bounded by a solid of revolution. For example: If I am given $z=x^2 $ which is a parabola, and ...
2
votes
0answers
116 views

Fubini theorem for improper Riemann integral

Is there a version of Fubini's theorem for improper Riemann integrals? Here's an example of what such a version might look like. If $f:\mathbb{R}^n\rightarrow\mathbb{R}$ is bounded and non-negative ...
0
votes
3answers
81 views

Standard Triple Integral Problem

Evaluate $ \iiint_D (x^2+y^2) \, ,\mathrm{d}V $, where $D$ is the region bounded by the graphs of $y=x^2$, $z=4-y$, and $z=0$. How would I do this problem? I can't even visualize the region D. ...
4
votes
2answers
131 views

Show that there is $(a,b)$ s.t. $f_x(a,b)^2+f_y(a,b)^2<4$

Let $f:D\to\mathbb{R}$ where $D:=\{(x,y):x^2+y^2\leq 1\}$. Suppose all partial derivatives of $f$ are continuous and $|f|\leq 1$. Show that there exists $(a,b)\in Int_D$ s.t.: $$\left[\frac{\partial ...
1
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2answers
36 views

Does $g(x,y,z)$ (the equation of the surface) need positive $z$ or negative $z$ when doing a surface integral?

$\quad$If a smooth surface $S$ is defined by $g(x,y,z)=0$, then recall that a unit normal is $$\mathbf{n}=\dfrac{1}{\|\nabla g\|}\nabla g,\tag{9}$$ where $\nabla g=\dfrac{\partial g}{\partial ...
2
votes
2answers
639 views

Why gradient vector is perpendicular to the plane

I know what gradient vector or $\nabla F$ is and I know how to prove that it is orthogonal to the surface (using calculation - not intuitive). In a particular case, in which we have a three variable ...
1
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4answers
127 views

Marginal density function understanding

Given a plane with three points, $(0, -1)$, $(2,0)$, and $(0, 1)$ with $x$-axis and $y$-axis connecting three points to make a triangle. Suppose this triangle represents the support for a joint ...
0
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1answer
31 views

Plane tangent to both paraboloids

Can someone help me with this calculus question? It appeared on the past exams of my multivariable calculus class. Two paraboloids, one with equation $z=-1-x^2-y^2$ and the other with equation ...
0
votes
1answer
41 views

Computing the surface integral of a parabloid

Problem: Solution: I am having difficulty understanding how the author determined the limits of integration of $R$. The author used $\theta=\pi/3\quad to\quad \theta=\pi/2$ and $r=1\quad to\quad ...
0
votes
1answer
114 views

Integral of Gaussian ring/shell

I would like to know the integral of the function $$f(\mathbf{x}) = {1 \over \sqrt{2\pi \sigma^2}} \exp\left\{- {(|\mathbf{x}| - \mu)^2 \over 2\cdot \sigma^2}\right\} $$ over an $n$-dimensional ...
2
votes
1answer
169 views

Difference quotient (pde)

Let $u: U\subset\mathbb{R}^n\rightarrow\mathbb{R}$. The Difference quotient of $u$ is defined by $D_k^hu(x)=\dfrac{u(x+he_k)-u(x)}{h}$ with $h\in\mathbb{R}$, $0<|h|<\textrm{dist}(V,\partial ...
1
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1answer
65 views

On a system of PDE

I would like to know what is the set of solutions to the following PDE. I think it consists of just constants, but I need help to prove. Let $f_1(p_1,p_2)$ and $f_2(p_1,p_2)$ be two functions. The ...
2
votes
1answer
62 views

How to find the partial derivative of this function?

Lets say I have a function:$$\nu=\frac{RT}{P}+B_{p}(T)RT$$ and I am trying to find $\left(\frac{\partial \nu}{\partial T}\right)_{P}$. I understanding that the partial derivative of the first term is ...
0
votes
1answer
43 views

Help setting up integral

Let $A$ be the region in $\mathbb{R}^3$ bounded by the planes $x=0$, $y=0$, $z=2$, and the surfance $z=x^2+y^2$. Evaluate $$\int_A x\, \mathrm{d}x\, \mathrm{d}y\, \mathrm{d}z$$ Here's what I have but ...
2
votes
3answers
181 views

Simple criteria for “closed $\Longrightarrow$ exact”

In determining whether a closed form is an exact form, there is a lot of differential geometry definitions etc. that come in. I'm interested: what is the dummy, Calc III version of when closed implies ...
0
votes
1answer
85 views

Multivariable Calculus, Two Path Test.

I'm have trouble understanding how you determine which paths you should choose. In the book for the function lim as (x,y) goes to (0,0) $$f(x,y) = \frac{2x^2y}{x^4+y^2}$$ They say "We examine the ...
0
votes
1answer
58 views

Another exercise from Fleming's Functions of Several Variables.

I'm using Flemming's book Function of Several Variables. In it, the author defines Manifolds like this: Let $1\le r\lt n,\ q\ge1$. A nonempty set $M \subset \mathbb{R}^n$ is a manifold of dimension ...
0
votes
1answer
54 views

Intersection of a manifold with open set

I'm using Flemming's book Function of Several Variables. In it, the author defines Manifolds like this: Let $1\le r\lt n,\ q\ge1$. A nonempty set $M \subset \mathbb{R}^n$ is a manifold of dimension ...
19
votes
1answer
550 views

How prove there exists a point $(x_{0},y_{0})$, such $\Delta f|_{(x_{0},y_{0})}\ge 0$

Question: Assume that the function $f(x,y)$ is twice continuously differentiable on $\mathbb R^2$, and $$f\big|_{\partial\Sigma}=0,\quad \text{where}\,\,\,\partial\Sigma=\{(x,y)\in\mathbb ...
1
vote
0answers
92 views

Motion in three dimensions with friction.

I am trying to represent the motion of an object in three-dimensional space that is undergoing acceleration, friction, and drag, where: acceleration = $\vec A$ friction = $F$ drag = $D$ The ...