Tagged Questions

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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2
votes
2answers
1k views

Prove a cubic equation has at least one real root

Show that the cubic eq: $$x^3+ax^2+bx+c = 0 \quad a,b,c\in \mathbb{R}$$ has at least one real root. I know that the above equation can be broken down into $(x-a)(x-b)(x-c) = 0$ , but I have no ...
2
votes
1answer
28 views

sufficient condition for being an integral factor

Let $ f: \mathbb {R}^m \rightarrow \mathbb {R}-\{0\} $ function $C^{\infty}$ class and $w$ a one-form $C^{\infty}$ class in $\mathbb {R}^m $. If $\alpha=w-\dfrac{1}{f}dx_{m+1} $ satisfies $\alpha ...
0
votes
2answers
71 views

Second order quasilinear PDE

Some quick question about PDE's. Only recently started studying PDE's so this might be trivial. The second-order quasilinear elliptic equation is given by: $ -\sum_{i=1}^{n} \frac{\partial}{\partial ...
3
votes
0answers
115 views

Knowing when to use Green/Stokes/Divergence theorem to evaluate line/surface integrals

$\newcommand{\mbf}{\mathbf}$ Evaluate $$ \iint \limits_{S} \mbf{F} \cdot d \mbf{S} $$ where $\mbf{F} = 3xy^2 \mbf{i} + 3x^2y \mbf{j} + z^3 \mbf{k}$ and $S$ is the surface of the unit sphere. I ...
2
votes
1answer
111 views

Find partial derivative at point $(0,0)$ of $(xy)/(x^2+y^2)$

It's a bit wierd question but I have to ask it. $$ \text{Let }\space f(x, y) = \begin{cases} \dfrac{xy}{x^2 + y^2}, & \text{if $(x, y) \ne (0,0)$} \\ 0, & \text{if $(x, y) = (0, 0)$} \\ ...
1
vote
3answers
53 views

How to find this partial derivative?

So I have $z=x^2+xy+y^2$ And I want derivative of z with respect to x assuming y is constant and professor gave us $\frac{\partial z}{\partial x}=2x+y$ But how does he found it? Does he use limit ...
1
vote
2answers
46 views

Finding the tangent plane

Find the tangent plane to $$z=4 x^3+3 xy +4 y^3$$ at $(-1,1,-3)$. Answer on the form $z=Ax+By+C$. I don't know how to solve these problems, should I find the derivative of x and y, then let those be ...
0
votes
1answer
38 views

limit of the triple integral

I found $\iiint_E$ $1\over (x^2+y^2+z^2)^{n/2}$ dV, where E is the region bounded by the spheres with radiuses r and R (both positive), is $4\pi\left(\frac{R^{3-n}}{3-n}-\frac{r^{3-n}}{3-n}\right)$. ...
1
vote
1answer
39 views

How to find $\lim_{(x,y)\to(1,1)} \frac{x^3-y^3}{x^2 - y^2}$?

Suppose $$ f(x,y) = \begin{cases} \frac{x^3-y^3}{x^2 - y^2}, & \text{if $x^2 \ne y^2$} \\ \alpha, & \text{if $x^2 = y^2$} \\ \end{cases}$$ I need to find $\alpha$ such that $f(x,y)$ will be ...
1
vote
1answer
33 views

What happens at the point whose differential is nonzero

The question is: Suppose that $f$ is differentiable in $U = D(0;1)\subset \mathbb{R}^2$ and the Jacobian matrix of $f$ at $(0,0)$ is nonzero. Is it true that $f(x,y) \neq f(0,0)$ in some deleted ...
0
votes
2answers
59 views

Finding the directional derivative of f(x,y).

Let the directional derivative of a function $f(x,y)$ at a point $P$ in the direction of $(1/\sqrt{5})\mathbf{i}+(2/\sqrt{5})\mathbf{j}$ be $16/\sqrt{5}$ and the partial derivative $\partial f / ...
1
vote
1answer
105 views

Proving a scalar function is differentiable at the origin but that its partial derivatives are not continuous at that point.

Let $f: \mathbb R^2 \to \mathbb R$ defined as $f(x,y)=(x^2+y^2)\sin(\dfrac{1}{\sqrt{x^2+y^2}})$ if $(x,y) \neq (0,0)$, $f(x,y)=(0,0)$ if $(x,y)=(0,0)$ Prove that $f$ is differentiable at $(0,0)$ ...
1
vote
2answers
32 views

Approach to the integral $\int d^3u \exp(-\alpha|\mathbf{w}-\mathbf{u} |^2)\delta(\mathbf{k}\cdot\mathbf{u})$

I am trying to evaluate the integral $\int d^3u \exp(-\alpha|\mathbf{w}-\mathbf{u} |^2)\delta(\mathbf{k}\cdot\mathbf{u})$ where I have 3 vectors w,u, and k, a constant alpha, and the integral is ...
0
votes
1answer
80 views

Help with solving for a flow curve:

So I'm preparing for a final exam in multivariable and our textbook posed the following question: find the flow lines of F(x,y) = (-y, x) Which I can't seem to solve correctly. We are told that a ...
1
vote
1answer
65 views

Does there exist a $\nabla$-notation variant of the product rule applied to $\nabla[\mathbf{f}(\mathbf{x})\otimes\mathbf{g}(\mathbf{x})]$?

This is a vector-calculus notation question; as a disclaimer, I am working in rectilinear space! For vector functions $\mathbf{f},\mathbf{g}:\mathbb{R}^n\rightarrow\mathbb{R}^n$, the chain rule for ...
2
votes
4answers
4k views

Use implicit differentiation to find dy/dx

$xy+x=2$ I know the answer is $-(1+y)\over x$, but I don't know how to solve to get the answer. Thank you!
0
votes
1answer
18 views

Suggestions on how to compute $\int d^3 u \frac{1}{u}\exp(-(\textbf{w}-\textbf{u})^2/c^2)$?

I was looking for which coordinates and change of variables would make the integration of $\int d^3 u \frac{1}{u}\exp(-(\textbf{w}-\textbf{u})^2/c^2)$ the easiest. Note that the bold faced values are ...
2
votes
2answers
907 views

Volume of a sphere (r=2a) with hole(r=a) drilled through centre, using spherical polar coordinates.

Need help solving 11.bi), A cylindrical hole of radius a is bored through the center of a sphere of radius 2a. Find the volume of the remaining material, using spherical polar coordinates. (You ...
1
vote
2answers
144 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
votes
4answers
128 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
votes
1answer
56 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 ...
0
votes
1answer
62 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
27 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
votes
1answer
72 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
votes
1answer
91 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
64 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
vote
1answer
98 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
vote
1answer
46 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
342 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
65 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
60 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
34 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
vote
2answers
73 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: ...
2
votes
2answers
280 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
280 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
126 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
35 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
140 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
59 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
239 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
votes
1answer
128 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
82 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
2k 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
51 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
50 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
vote
1answer
36 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
100 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
49 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
91 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 ...