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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0
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1answer
34 views

Show that $ f$ is strongly differentiable at $x_0$ .

Definition: Let $U\subseteq \Bbb R^m$ be an open set. Let $f: U \to \Bbb R^n$ be a function and $T: \Bbb R^m \to \Bbb R^n$ be a linear transformation. We say that f is strongly differentiable ...
0
votes
1answer
51 views

Expectation of a linear combinations of iid standard normal, restricted to a halfspace

Let $u = (u_1, \ldots, u_n)\in\mathbb{R}^n$ be a unit vector in $\mathbb{R}^n$, $Y_i$ be i.i.d standard normal Is there any easy way to calculate $$\mathbb{E} \left[ 1_{\displaystyle \left\{ ...
3
votes
4answers
105 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 $$\lim_{(x,y,z)\to(0,0,0)}\frac{xy+xz+yz}{x^2+y^2+z^2}$$
1
vote
0answers
12 views

Convex set of derivatives implies mean value theorem

Let U$ \subset$ $R^{^{n}}\ $be open, $f:U\rightarrow R^{m}$ differentiable on U, and segment $[a,b]\subset U$. Assume that the set of derivatives { f'(x)$\in L(R^{^{n}},R^{^{m}}):x\in$ [a,b] }is ...
0
votes
0answers
31 views

divergence and curl of the function $(x^2+y^2)\log(1-z)$

I have been given the function $F(x,y,z) = (x^2+y^2)\log(1-z)$ and I need to find the divergence and curl. I understand that $$\nabla \cdot F = \frac{\partial F_x}{\partial x} + \frac{\partial ...
0
votes
0answers
13 views

Orthogonal surfaces

Prove that the three surfaces of the family $xy/z=u$ $\sqrt{x^2+y^2}+\sqrt{y^2+z^2}=v$, $\sqrt{x^2+y^2}-\sqrt{y^2+z^2}=w$ that pass through just one point are orthogonal I´m assuming that first I ...
0
votes
2answers
43 views

Differentiability and continuity at the origin of piecewise defined $g(x,y) = y-x^2$, $y+x^2$, or $0$

$$g(x,y)= \begin{cases} y-x^2, & y\ge x^2\\ y+x^2, & y\le -x^2\\ 0 & \text-x^2\le y\le x^2 \end{cases}$$ I need to find all the directional derivatives at the origin in the tangent ...
1
vote
2answers
7k views

Derivative of cross-product of two vectors

In finding the derivative of the cross product of two vectors $\frac{d}{dt}[\vec{u(t)}\times \vec{v(t)}]$, is it possible to find the cross-product of the two vectors first before differentiating?
0
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0answers
25 views

Numerical Triple integral with three other parameters in R

I am trying to do this triple integral $$\int_{0}^{\infty }\int_{0}^{\infty }\int_{0}^{\infty }(u+w)e^{-\frac{(u+w)^2}{2}}(v+w)e^{-\frac{(v+w)^2}{2}}(u+v)e^{-\frac{(u+v)^2}{2}}e^{-(\mu +\lambda ...
0
votes
1answer
15 views

Rate of Change of a Multivariable Function

The problem says, Find the rate of change of $$(x,y,z) = x/z + y/z$$ with respect to t along the curve $$r(t) = \sin^2{t}[ i] + \cos^2{t}[j] + 1/(2t)[k]$$ The answer is apparently ...
4
votes
2answers
40 views

How to calculate this multivariable limit?

$$ \lim_{(x,y,z)\to (0,0,0) } \frac{\sin(x^2+y^2+z^2) + \tan(x+y+z) }{|x|+|y|+|z|} $$ I know the entire limit should not exist. In addition, the limit: $$ \lim_{(x,y,z)\to (0,0,0) } \frac{\tan(x+y+z) ...
1
vote
1answer
19 views

Smooth manifold

$A=M\cap N$, $M={(x,y,z\in\Bbb R^3)| x^2+y^2=1}$, $N=(x,y,z)\in \Bbb R^3|x^2-xy+y^2-z=1$. $1$. Is $A$ is smooth manifold? $2$. Find the points of $A$ that are farthest from the origin. This is ...
6
votes
1answer
203 views

Equivalence of 2 definitions of Differentiability

Let $X,Y$ be Banach spaces. I would like to prove the equivalence of the following definitions of differentiability. Let $f:X\to Y$ and $a\in X$ There is a map $\Delta : X \to L(X,Y)$ continuous at ...
1
vote
1answer
18 views

Center of mass and moment of inertia of a $2$-dimensional donut?

I have a an assignment and I'm stuck on this question: First of all I can't figure out the equation for a $2$-dimensional donut as shown in the diagram. For the calculation of the center of mass, I ...
0
votes
4answers
46 views

Remember the implicit function theorem

First, I know the implicit function theorem, but unfortunately I always have to look it up again and again. If $F(x,y)=0$ then I always forget whether I have to invert the first matrix of the Jacobian ...
0
votes
0answers
19 views

Find global minimum of the function

I need to find the global minimum of the function $$f ( x) = \langle Ax,x \rangle + 2\langle b ,x\rangle+c$$ where $c \in \mathbb{R}$ is constant, $b \in \mathbb {R}^n$, and $A$ is a positive ...
0
votes
1answer
17 views

Calculate the level set of $(x^2+y^2)\log(1-z)$

I need help with the following question if possible. I'm not entirely sure on how I should begin. Find the level set of $f$ which has value (“height”) $w_0 = 0$, and describe it in words and set ...
0
votes
0answers
12 views

Computing the index around a curve with respect to a field, invariance?

If I understood the course book Nonlinear Dynamics and Chaos right, The index can be found by $$\newcommand{\dd}{\mathrm{d}} \newcommand{\id}{\mathrm{d\,}} I_{C}=\frac{1}{2\pi}\oint_C ...
2
votes
0answers
51 views

Another proof of Inverse Function theorem in $\mathbb{R}$

(Inverse Function theorem in $\mathbb{R}$) Suppose $I\subset \mathbb{R}$ is an open interval and $f:I\rightarrow\mathbb{R}$ is a differentiable function.If for all $x\in I$ is such that $f^{'}(x)\ne ...
0
votes
1answer
18 views

Calculate the taylor polynomial $f(x,y,z) = (x^2+y^2)log(1-z)$

I have been given the following question that I need help with. I have calculated grad f but I'm just not sure exactly how to calculate the taylor polynomial which is part c. Consider the function ...
0
votes
1answer
59 views

If $f(U)=0$ then what is possible?

Let , $U=\left(0,\frac{1}{2}\right)\times \left(0,\frac{1}{2}\right)$ and $V=\left(-\frac{1}{2},0\right)\times \left(-\frac{1}{2},0\right)$ and $D$ be the open unit disk centered at origin of $\mathbb ...
2
votes
1answer
33 views

Area under the curve described by θ=ar

I'm interested in finding the area under the curve described by θ=ar, which is a linear curve with slope 'a' in polar coordinates. Here is what the curve looks like: ...
0
votes
1answer
38 views

Explicitly find all pairs $(a,b)$ such that $a^{1/a}=b^{1/b}$ and $a\ne b$.

Explicitly find all pairs $(a,b)$ s.t. $a^{1/a}=b^{1/b}$ and $a\ne b$. My multivariable calculus teacher posed this question as a fun brain teaser for the end of the semester. He said it was ...
1
vote
1answer
34 views

Double integral of $\arctan(x + y)$?

I would like to find $\int_a^b\int_a^b\arctan(x+y)dydx$ I can "simplify" the integration down to $\int_a^b ((x+b)\arctan(x+b)-\frac{1}{2}\ln(1+(x+b)^2) - ...
1
vote
1answer
25 views

Convergence rate - Convex optimization

What is the best known algorithm in terms of convergence rate for unconstrained convex optimization and under what assumptions? $\min_{x} f(x)$ where $f(x)$ is a given twice differentiable convex ...
0
votes
1answer
25 views

Compute the wedge product n times

Let $\omega$ be a 2-differential form in $\mathbb{R}^{2n}$ given by $$\displaystyle \omega=dx^1\wedge dx^2+dx^3\wedge dx^4 + \cdots + dx^{2n-1}\wedge dx^{2n}$$ Compute: $$\displaystyle ...
-2
votes
1answer
43 views

How can I integrate $\sqrt{1-x^2/a^2-y^2/b^2} dx dy$ in the ellipse? [on hold]

I am lookig for $$\int\int _{D} \sqrt{1-\frac{x^2}{a^2}-\frac{y^2}{b^2}} dxdy $$ where $D$ is defined by $D=\{ (x,y) \in \mathbb{R} \mid \frac{x^2}{a^2} + \frac{y^2}{b^2}\leq 1 \}$. please help
1
vote
1answer
31 views

Some conditions on $\tilde f(x,y)=\begin{cases}\displaystyle g(x,y) & \text{if }(x,y)\not=(0,0), \\ 0 & \text{if } (x,y)=(0,0).\end{cases}$

The following function $$f(x,y)=\begin{cases}\displaystyle\frac{x^2 y^2}{x^2+y^2} & \text{if }(x,y)\not=(0,0), \\ 0 & \text{if } (x,y)=(0,0).\end{cases}$$ is differentiable in the origin and ...
2
votes
1answer
32 views

An Application of Stokes's Theorem

Let $D^2=\{(x,y)\in \mathbf R^2: x^2+y^2\leq 1\}$ be the unit disc in $\mathbf R^2$, and $D^3=\{(x,y,z)\in \mathbf R^3: x^2+y^2+y^2\leq 1\}$ be the unit disc in $\mathbf R^3$. Let $i_{\pm}:D^2\to ...
-3
votes
2answers
42 views

Differentiability at zero [on hold]

Is the following function differentiable at $(0,0)$? $$f(x,y)=\begin{cases}\displaystyle\frac{x^3 y}{x^4+y^2} & \text{if }(x,y)\not=(0,0), \\ 0 & \text{if } (x,y)=(0,0).\end{cases}$$
0
votes
2answers
16 views

Converting rectangular coordinates to cylindrical coordinates and then integrating

$$\int_0^2 \int_9^\sqrt{2x-x^2} (xy) \ dy\ dx$$ I have to solve this problem by converting from rectangular coordinates to cylindrical coordinates then integrate it. I know that $$\ r^2 = x^2 + y^2 ...
3
votes
1answer
22 views

Why isn't $d\mathbf{A}$ normalized in Stokes' theorem?

For a nice curve $C$ which is a boundary of a smooth surface $D$, Stokes' theorem says that $$\begin{align*} \oint_C \mathbf{F}\cdot d \mathbf{s} = \iint_D (\nabla \times \mathbf{F} )\cdot ...
1
vote
1answer
29 views

Intersection of Eigenvectors and Multivariable Calculus

This isn't really a problem but more of a reference/example question: do eigenvalues and eigenvectors ever show up in multivariable calculus? The two seem very unrelated to me. Specific examples would ...
2
votes
1answer
27 views

Show that an open subset $U \subset \mathbb{H^n}$ is open in $\mathbb{R^n}$ iff $U \cap \partial\mathbb{H^n}=\phi$

Consider the $n-$dimensional closed half-space $\mathbb{H^n}=\{x \in \mathbb{R^n}|x_1 \le 0 \}$ and let $\partial \mathbb{H^n}=\{x \in \mathbb{H}|x_1=0\}$ be its boundary. Show that an open subset $U ...
0
votes
1answer
36 views

Derivatives - Show equality

Let $y(x)$ be defined implicitly by $G(x,y(x))=0$, where $G$ is a given two-variable function. Show that if $y(x)$ and $G$ are differentiable, then ...
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votes
0answers
18 views

Gaussian Curvature. [on hold]

Show that the Gaussian curvature of the surface at $(x_0,y_0)$ is equal to the discriminant $D$ appearing in the second derivative test at $(x_0,y_0)$. $(x_0,y_0)$ is a critical point and $r(x,y)= ...
1
vote
0answers
21 views

Difficult examples of invertible, differentiable functions

Give an example of: 1)$f:\mathbb R^2 \to \mathbb R^2$ such that $f$ is invertible in some neighbourhood of $x_0$ (that is $f$ is locally invertible) but $|Jf(x)|=0$ (jacobian determinant) $\forall x$ ...
1
vote
2answers
23 views

What is the integral of this function over a cylinder?

What is the $\iint_G x^2 z $ where $G$ is the closed cylinder $x^2 + y^2 = 4$, and $0 \le z \le 3$. I parametrized the outside of the cylinder as $G(\theta,z)=(2\cos\theta,2\sin\theta,z)$ , with $0 ...
0
votes
1answer
30 views

Show that the function is differentiable

I have to prove that the following function is differentiable and to find its derivatives at any point. $$f: \mathbb{R}^2 \rightarrow \mathbb{R}, (x,y) \rightarrow x^2+y^2$$ In my book there is a ...
1
vote
1answer
30 views

Convservation of Momentum

I am taking a course in fluid dynamics. I'm trying to establish the equality $$\frac{d}{dt}\int_{a(t)}^{b(t)} \rho(x, t)g(x, t)dx = \int_{a(t)}^{b(t)}\rho(x, t)\frac{D}{Dt}g(x, t)dx$$ I can use make ...
2
votes
1answer
9 views

Multivariable Calculus: Line Integrals (Directed Curve)

I have this math problem, that I got a bit confused on. I just need to know whether or not I did it correctly. Thanks! Question: Calculate $\oint_c xe^{z}dx+yzdy+xe^{y}dz$ over the directed curve ...
4
votes
1answer
113 views

Multiple integral differential notation

When writing a multiple integral, there is sometimes used a shorthand for writing the differential in the integral. For example in $\mathbb{R}^3$ instead of writing $\mathrm{d}x\ \mathrm{d}y\ ...
2
votes
5answers
37 views

Multivariable Calculus: Line Integral

I have this math problem. It states: Calculate the given line integral $\oint _c {M dx+Ndy}$ where $C$ is the triangle with vertices $P_0=(0, 1)$, $P_1=(2, 1)$, $P_2=(3, 4)$ with ...
0
votes
1answer
12 views

Providing an independent proof for change of variables

Given a smooth change of variables $\psi:\mathbb R^2\to\mathbb R^2$ where $\psi(x,y) = (x, g(x,y))$ and $g$ is strictly increasing with respect to its second variable. For $K = [a,b]\times[0,1]$, ...
1
vote
0answers
9 views

In what direction starting at $(1,2)$ does f increase most rapidly? What is the magnitude of this increase?

Find the directional derivative of $f(x,y)=\frac{y^2}{x}$ at $(1,2)$ in the direction of $v=<1,-1>$ i started by finding the gradient of f(x,y) which is $fx(x,y)=\frac{-y^2}{x^2}$ ...
0
votes
0answers
7 views

Moment of inertia of lower dimensional bodies and limits

I have noticed that the moment of inertia of an $n$-dimensional body having uniform density $\rho$ is at least sometimes identical to the limit of the moment of inertia of a $(n+1)$-dimensional body ...
1
vote
0answers
17 views

Find a vector function that represents the curve of intersection of $x^2+y^2=4$ and $z=x$

Find a vector function that represents the curve of intersection of $x^2+y^2=4$ and $z=x$ I changed $x^2+y^2=4$ to $4sin^2\theta + 4cos^2\theta = 4$ so $x=2cos\theta$ and $y=2sin\theta$ and then ...
1
vote
3answers
15 views

Converting unit square domain in (x,y) to polar coordinates

I have the following double integral $\int_{0}^{1}\int_{0}^{1}\frac{x}{\sqrt{x^2+y^2}}dxdy$ The integrand is fairly simple: $\frac{x}{\sqrt{x^2+y^2}}dxdy=\frac{rcos(\theta ...
1
vote
1answer
29 views

implicit multivariable derivative

I didn't really understand how implicit multi-variable functions are derived; I thought of another method which may fit and may not; suppose we have $xy^2z^3=8$ and we want to derive it; it is the ...
2
votes
0answers
13 views

uniqueness of solution of a nonlinear equation system

$f(x,y):\mathbb{R}_+\times \mathbb{R}_+\rightarrow\mathbb{R}_+$, is a differentiable function with partial derivatives $f_1(x,y)<f_2(x,y)<0$. $z_1,z_2,...z_n$ are n given positive numbers. ...