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
21 views

Fundamental theorem of calculus in multivariable calculus

I'm not sure if this is the right name for it but with the theorem: Let $f:\sigma \rightarrow \mathbb{R} $ be a smooth scalar field and assume $r: [a,b] \rightarrow \mathbb {R}^n$ is a piecewise ...
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2answers
29 views

fifth order Differential EQuation

find the general solution of higher order linear differential Equation? find the general solution of Differential equation using auxiliary equation? $2y^{(5)}-7y^{(4)}+12y"'+8y"$=0
1
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1answer
23 views

Calculating line integrals

The curve $\gamma$ is parameterized by: $t \rightarrow(\cos t, \sin t), t \in [0,2\pi]$ I want to calculate the following integrals and I am supposed to explain what "type" of integral each one is. ...
2
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2answers
33 views

Need help solving complicated integral $\oint_{\mathcal C}\begin{pmatrix}x_2^2 \cos x_1 \\ 2x_2(1+\sin x_1)\end{pmatrix} dx$

Let $\mathcal C$ be the curve that traces the unit circle once (counterclockwise) in $\mathbb R^2$. The starting- and endpoint is (1,0). I need to figure out a parameterization for $\mathcal C$ and ...
0
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0answers
16 views

Integrating 2 form over torus

Let $\Bbb M^2 ⊂ \Bbb R^3$ be the torus of revolution obtained by rotating the circle $(x−2)^2 +z^2 = 1$ in the $xz$ plane around the $z$ axis. Consider the orientation on $M$ induced by the ...
2
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2answers
23 views

Evaluating the triple integral $\iiint \limits_R ze^{-(x^2+y^2+z^2)} \, \, dV$

Evaluate the following triple integrals as a repeated integral using an appropriate coordinate systems: $$\iiint\limits_R ze^{-(x^2+y^2+z^2)} \, \, dV ,$$ where $$R=\{ (x,y,z): \, x,y \in (-\infty, ...
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0answers
10 views

Limit of a matrix valued function

The limit of a vector valued function $f:\mathbb{R} \to \mathbb{R^n}$ is defined as: $$\lim_{x \to a} f(x)=(\lim_{x \to a}f_1(x),\dots,\lim_{x \to a}f_n(x))$$ provided that the limits of $f_i(x)$ ...
4
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0answers
37 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 $\{ f'(x)\in L(R^{^{n}},R^{^{m}}):x\in [a,b] \}$ ...
0
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0answers
34 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 ...
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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
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2answers
49 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
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1answer
20 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 ...
4
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2answers
44 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
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1answer
24 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 ...
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4answers
47 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
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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
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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 ...
0
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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 ...
2
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0answers
52 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 ...
2
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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
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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 ...
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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) - ...
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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 ...
1
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1answer
27 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 ...
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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
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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
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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 ...
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2answers
42 views

Differentiability at zero [on hold]

Is the following function differentiable at $(0,0)$? $$f(x,y)=\begin{cases}\displaystyle\frac{x^3 y}{x^4+y^2} & \text{if }(x,y)\not=(0,0), \\ 0 & \text{if } (x,y)=(0,0).\end{cases}$$
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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 ...
0
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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 ...
3
votes
1answer
24 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 ...
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0answers
20 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)= ...
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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
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1answer
30 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
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1answer
10 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 ...
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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
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1answer
31 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 ...
2
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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 ...
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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 $K = [a,b]\times[0,1]$, ...
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0answers
10 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 $fx(x,y)=\frac{-y^2}{x^2}$ ...
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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 ...
0
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1answer
37 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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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 ...
2
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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. ...
0
votes
0answers
20 views

Finding this second order partial derivative

Want to find the second order of $$\frac{\partial z}{\partial x}=\frac{\partial z}{\partial g}\frac{\partial g}{\partial x}+\frac{\partial z}{\partial h}\frac{\partial h}{\partial x}$$ Normally if I ...
1
vote
1answer
15 views

Find the length of the curve r(t)= <t^2,2t,lnt> from t=1 to t=e

Find the length of the curve r(t)= $<t^2,2t,lnt> $ from t=1 to t=e i know that Length= $\int$ length of r'(t) dt Therefore, L= $\int _1^e\sqrt{4t^2+4+\frac{1}{t^2}}dt\$$ but i'm having trouble ...
1
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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
30 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 ...
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0answers
17 views

Integral identity of product of three pair of points.

I came across an identity could someone please help me deriving it ? $$ \int d^3t|t-x|^a |t-y|^b |t-z|^c= \pi^{3/2} C(a) C(b) C(c) |x-y|^{-3-c} |y-z|^{-3-a} |z-x|^{-3-b} $$ where $a+b+c = -6$ The ...