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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31 views

Stokes theorem on oriented curve of form $(x+y+z)dx + x^2dy+xyzdz$

This is the formulation, which is not clear to me entirely. Let $S$ be the upper unit half-sphere (this probably means the set $\{(x,y,z)|x^2+y^2+z^2=1, z\geq 0\}$) in the right half-plane plane(I ...
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0answers
17 views

Find the area of the surface of the sphere $ x^2 + y^2+z^2= a^2$ which is inside the cylinder $x^2+y^2 = ay.$

$ z = \sqrt{(a^2-x^2-y^2)};\; \frac{\partial z}{\partial x} = \frac{-x}{z}\; \frac{\partial z}{\partial y} = \frac{-y}{z};$ $\sqrt{1+\frac{x^2}{z^2}+\frac{y^2}{z^2}} = \frac{a}{z} = \frac{a}{\sqrt{a^2-...
2
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1answer
24 views

Finding the volume of Torus, Jacobian of spherical substitution.

I thought to find the volume of a Torus, like I would a sphere, where the spherical substitution was: $$x=r\cos\varphi\sin \theta , y= r\sin\varphi \sin \theta, z=r\cos \theta \\ g(r,\varphi,\theta)\...
-7
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0answers
66 views

Triple integral tetrahedron [on hold]

I need to find the volume of the region in the $xy$-plain bounded by: The coordinate planes The plane $x+y=4$ The plane $y+z=4$ And I am told to do so by using the triple integral in the order $...
1
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1answer
16 views

Condition for an expression to be a total differential

I have fully understood the concept and formulae around total differentials of multivariate functions. What is the condition however for an expression of differentials to be the total differential of ...
1
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2answers
28 views

Find stationary points of the function $f(x,y) = (y^2-x^4)(x^2+y^2-20)$

I have problem in finding some of the stationary points of the function above. I proceeded in this way: the gradient of the function is: $$ \nabla f = \left( xy^2-3x^5-2x^3y^2+40x^3 ; x^2y+2y^3-x^4y-...
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0answers
20 views

Show that $\int_{\Bbb{R}^n}f=nV\int_{0}^{\infty}g(r)r^{n-1}dr$ for $f(x)=g(||x||)$ and $g:[0,\infty)\to [0,\infty)$ integrable

Let $||\cdot||$ be the $p$-norm with $p>1$, and let $g:[0,\infty)\to[0,\infty)$. Let $f:\Bbb{R}^n\to [0,\infty)$ be defined by $f(x)=g(||x||)$. Show that if $g$ is integrable, so is $f$, and that $...
1
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1answer
49 views

Proof of a famous theorem in multivariable calculus

How do I prove the following famous theorem using simple calculus: $\int\dfrac{d}{dt}f(t,x)dx=\dfrac{d}{dt}\int f(t,x)dx$ or how is it that by first differentiating and then integrating a multi-...
1
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2answers
28 views

Find volume of cone in sphere

I can't understand why the projection of those two objects on plane $xy$ gives that the angle's range from -90 degress to 90 degrees. why not from 0 to 360? Edit: from my understanding, if $x > ...
3
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2answers
35 views

Conjecture about Cal 1 derivatives?

Conjecture: Let $F\left(\vec{x}\right) : \Bbb{R}^n \to \Bbb{R}$ Define $g(t) = F(t, t, \dots, t)$ Then $$g^{\prime} (t) = \left(\sum_{i=1}^n \ { \partial F \over \partial x_i}\right)\...
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0answers
15 views

Using Gauss Divergence Theorem Please help calculate

Please help me. I found div F. But I cannot handle the w.
0
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2answers
37 views

How do you find a vector function that represents the curve of intersection of a sphere and a plane?

sphere -> $x^2 + (y-\pi)^2 + (z - \frac{1}{18})^2 = (\frac{37}{18})^2$ plane -> $x + 6y = 6\pi$ Solving for y in the plane equation I get: $y = \pi - \frac{x}{6}$ Then plugging $y$ into the ...
0
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1answer
16 views

Find the equation to the tangent plane

$f(x,y) = \sqrt{xy}$ at the point (1,1,1) $f_x$(1,1) = $\frac{\sqrt{y}}{2\sqrt{x}}$ = $\frac{1}{2}$ same for $f_y$ setting up the formula I get: $\frac{1}{2}$$(x-1)$+$\frac{1}{2}$$(y-1)$$=$$z-1$ ...
0
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0answers
12 views

Let $Q=[0,1]\times[0,2]$. Find the rotation matrix $(A)$for the angle $\frac{\pi}{4}$ upon this set. Is $A(Q)$ measurable and a elementary set?

Let $Q=[0,1]\times[0,2]$. Find the rotation matrix $(A)$for the angle $\frac{\pi}{4}$ upon this set. Is $A(Q)$ measurable and a elementary set? First off, I know that $A$ is a linear map, and a ...
2
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2answers
37 views

Find the partial derivative of $\arctan (x/\sqrt{x^2+y^2})$ using the definition [closed]

Let $f(x,y)=\arctan \frac{x}{\sqrt{x^2+y^2}}$. How to evaluate $$\lim_{h\to 0}\frac{f(4h,1)-f(h,1)}{h}?$$
4
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1answer
82 views

PDF of the difference between two independent beta random variables

I am having trouble deriving the distribution of the difference of two beta random variables and would like some help verifying the steps I have taken. In particular calculating the bounds. Say I ...
0
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2answers
44 views

Generalities regarding the Lagrange Multiplier

Apparently the following general statement is true. "Let $\gamma:g(x,y)=0$ be a closed curve that doesn't cross itself. If the maximisation of a function $f(x,y)$ on $g(x,y)$ using Lagrange ...
1
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2answers
47 views

How to interpret $\nabla g(x,y)\cdot\begin{pmatrix}x\\y\end{pmatrix}<0$

Let $g\in C_1(\mathbb{R}^2,\mathbb{R})$. Suppose that $$ \nabla g(x,y)\cdot\begin{pmatrix}x\\y\end{pmatrix}<0 $$ for all $x,y\in\mathbb{R}$ with $x^2+y^2=1$. Show that there exist $x_*,...
0
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1answer
34 views

Solutions of a system of polynomial equations

I am trying to find the critical points of some functions such as $$f(x, y) = x^4 − x^2y^2 + y^3 − 18x^2 + 3y^2$$ I calculate the gradient, and then find a system of polynomial equations: $$\...
0
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2answers
28 views

Show that the iterations $x_{n+1}=x_n+y-f(x_n)$ converge to a solution of $f(x)=y$ for every $x_0,y\in \Bbb{R}^n$

Let $f:\Bbb{R}^n\to \Bbb{R}^n$ satisfy $$\forall x_1,x_2\in \Bbb{R}^n, |(f(x_1)-f(x_2))-(x_1-x_2)|\le {1\over 2} |x_1-x_2|$$ Show that the iterations $x_{n+1}=x_n+y-f(x_n)$ converge to a solution of $...
2
votes
1answer
25 views

Finding minimizer from different order

Let a nonnegative function $f(x,y)$: $\mathbb R^2\to \mathbb R$ be second order continuous differentiable. We also know that $f$ is not convex in its two arguments, but only separately in each of them....
0
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0answers
35 views

Compute the area of that part of the surface of the sphere $x^2 + y^2 + z^2 = a^2$ cut out by the surface $\frac{x^2}{a^2}+\frac{y^2}{b^2} = 1$.

I need help, I'm studying the area of surfaces using double integrals, I attempted to solve this exercise but I'm having trouble. The problem is to compute the area of that part of the surface of the ...
0
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1answer
63 views

Does continuity imply partial derivatives exist?

I know that in functions of more than one variable, the existence of partial derivatives does not guarantee that the function will be continuous. However, can the reverse be stated? i.e. that the ...
1
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2answers
30 views

Integral over domain enclosed by curves

Integrate the following function over the domain $D$ which is limited by the curves $xy=1, xy=2, y-x=1, y-x=3.$ \begin{equation} \int\int_{D} (x+y)dxdy \end{equation} I have drawn out the domain and ...
2
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1answer
71 views

How do I calculate the integral $\int_{0}^{1}{\frac{xe^{ax}}{(1+ax)^2}dx}$? [closed]

If $a>0$, how do I calculate the following integral? $$\int_{0}^{1}{\frac{xe^{ax}}{(1+ax)^2}dx}$$
0
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1answer
15 views

Helmholtz Decomposition on $\mathbb{R}^3$ Proof

I am trying to prove the Helmholtz decomposition theorem which states that given a smooth vector field $\mathbf{F}$, there are a scalar field $\phi$ and a vector field $\mathbf{G}$ such that \begin{...
0
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0answers
21 views

Is the Inverse of a Coordinate System on a Differentiable Manifold Differentiable?

In his book Calculus on Manifolds, Spivak shows that a differentiable k-dimensional manifold in $\mathbb{R}^n$ is a set $M$ such that $\forall x \in M$ there exists an open set $U$ containing $x$, an ...
1
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1answer
46 views

Want to confirm that I got the right answer to an “iterated integrals” question.

For positive real numbers $R$ and $r$, let $$E(R, r) = \{\frac{x_1^2 + x_2^2 + x_3^2}{R^2} + \frac{x_4^2}{r^2}\leq 1\}$$ Using an iterated integral, calculate the volume of $E(R,r)$. I am not ...
0
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1answer
31 views

Does an unbounded gradient implies a non-vanishing hessian determinant

Let $\Phi:\mathbb{R^{n-1}\to R}$ be a vector function. Suppose that exists a set $S_1\subset \mathbb R^{n-1}$ on which $G=\nabla\Phi$, the gradient of the function is unbounded. Does that imply ...
1
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1answer
34 views

For $E\in \Bbb{R}^3$,$(x,y,z)\in E \iff (x,y,-z)\in E$, and linear $ f:\Bbb{R}^3\to \Bbb{R}$, if $(x_0,y_0,z_0)$ is the center of mass, $z_0=0$

Let $E\subset \Bbb{R}^3$ be a measurable set (i.e. $\int_{\Bbb{R}^n}1_{E}$ exists) and let $v(E)\ne 0$. Let $f$ be a linear function $f:\Bbb{R}^3\to \Bbb{R}$, and let $(x_0,y_0,z_0)$ be the center of ...
0
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2answers
68 views

How do I determine maximum/minimum of this multivariable function?

Determine maximum and minimum of function $f\left(x,y\right)=x^{3}+y^{3}$ on range $B=\left\{ \left(x,y\right): x^{2}+y^{2}\leq 1 \right\} $ Show that maximum and minimum can only be on this range ...
1
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1answer
45 views

Let $R$ be a rectangle and let $f$ bounded. Show that $f$ is integrable in $R$.

Let $R$ be a rectangle of $\mathbb{R}^n$ and let $f:R \to \mathbb{R}$ bounded. Assume that there exists a sequence $\{P_n\}$ of partitions of $R$ such that $$\lim_n [S(f,P_n)-s(f,P_n)]=0$$ Show ...
0
votes
1answer
80 views

Fréchet differentiability of $\frac{x^3y^2}{x^4+y^4}$ at $(0,0)$?

Suppose a function $f$ is defined as follows: $$f(x,y)=\begin{cases} \frac{x^3y^2}{x^4+y^4}&\text{ when }(x,y)\neq(0,0),\\0 & \text{ when }(x,y)=(0,0).\end{cases}$$ I want to determine ...
2
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0answers
36 views

Find $F'(x)$ where $F(t) := \int_{g(x)}^0 f(x,t)dt$

Let $f: \mathbb{R^2}\to \mathbb{R}$ be $C^1$ and $g:\mathbb{R}\to \mathbb{R}$ also $C^1$. Define $$F(t) := \int_{g(x)}^0 f(x,t)dt$$ Find $F'(x)$. What I did was first define $\Phi : \mathbb{R}\...
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2answers
32 views

vector and curl identity

This popped up in my notes and the author made no remarks about the properties used $\bigtriangledown \times \left ( \vec{E}+\frac{\partial \vec{A}}{\partial t} \right )=\vec{0}$ Then, $\...
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0answers
18 views

higher derivatives of $R^m \to R^n$ [duplicate]

What's a good source (paper, book, website,...) where I can learn more about higher derivatives of functions $R^m \to R^n$ as multilinear functions or tensors? Thanks.
0
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1answer
19 views

What is $(A_1 \times … \times A_n) \cup(B_1 \times … \times B_n)=?$ ,$A_i$'s are intervals

What is $(A_1 \times ... \times A_n) \cup(B_1 \times ... \times B_n)=?$ ,$A_i$'s are intervals $[a_{Ai},b_{Ai}]$ and $B_i$'s are $[a_{Bi},b_{Bi}]$ respectively. What I mean is can $(A_1 \times ... \...
1
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2answers
49 views

What is the measure of $A=[-1,2]\times[0,3]\times[-2,4]\cup[0,2]\times[1,4]\times[-1,4] \setminus [-1,1]^3$?

I really get stuck after one point, and don't know where to go on.I know that my try, up to where I am stuck is correct. $$\color{#20f}{\text{TRY:}}$$ $$B_1=[-1,2]\times[0,3]\times[-2,4],\mu(B_1)=3 \...
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1answer
34 views

Triply integral involving spherical coordinates - how can I proceed?

$$ \iiint_V \frac{1}{x^2+y^2 + z^2 } dx dy dz =? $$ where $$ V=\{ (x,y,z)| x^2 + y^2 + (z-1)^2 \leq 1 \}. $$ After moving to spherical coordinates I obtain: $$ \iiint \sin \theta dr d\theta d\phi $$ ...
2
votes
1answer
40 views

Let $A=\{(x,y,z)\in \mathbb{R}^2 : x^2+y^2+z^2 \leq 1, 0\leq z \leq \frac{1}{2} \}$. Find the volume of $A$.

Let $A=\{(x,y,z)\in \mathbb{R}^2 : x^2+y^2+z^2 \leq 1, 0\leq z \leq \frac{1}{2} \}$. Find the volume of $A$. The volume I'm asked to find it's what is left of the unit semisphere minus the upper ...
0
votes
2answers
29 views

Slope of the tangent of the curve $cos(x-y)+sin(x+y)=1$, when $x=y=\pi$

Now I have graphed this online and get Now by directly substituting, I get $cos(0)+sin(2\pi)=1$ which is just $1=1$. So what exactly is this question asking?
3
votes
3answers
124 views

Continuity of $\frac{x^3y^2}{x^4+y^4}$ at $(0,0)$? [duplicate]

Suppose a function $f$ is defined as follows: $$f(x,y)=\begin{cases} \frac{x^3y^2}{x^4+y^4}&\text{ when }(x,y)\neq(0,0),\\0 & \text{ when }(x,y)=(0,0).\end{cases}$$ Is this function ...
-1
votes
2answers
50 views

Find the tangent plane on $z=x^3-xy$ perpendicular to $(1,1,1)$

I'm not sure how to do this. I tried letting $\partial{z}/\partial{x}=1$ and $\partial{z}/\partial{y}=1$ then solving for $z$ at this point and subbing them into $x+y+z=c$ but I just get $x+y+z=0$
0
votes
2answers
52 views

How do I determine the maximum or the minimum of in the range the function in the range $B=\left \{ \left ( x,y \right ):x^2+y^2\leq 1 \right \}$?

How do I determine the maximum or the minimum of in the range the function $f(x,y)=x^3+y^3 $in the range $B=\left \{ \left ( x,y \right ):x^2+y^2\leq 1 \right \}$? As a continuous function f must ...
2
votes
1answer
40 views

Multidimensional taylor series $sin (x^3y^2) $

A homework of mine was to compute the Taylor series of $f(x,y)=\sin(x^3y^2)$ around $(0,0)$ to the 25th order. I assumed, as $\sin(z)=\sum\limits^{\infty}_{k=0}(-1)^k\frac{z^{2k+1}}{(2k+1)!}$, that I ...
1
vote
1answer
48 views

How to do a Taylor expansion of a vector-valued function

Let $f:\Bbb R^2\to \Bbb R^2$ be given by $$f(x,y):= \left(e^x\sin(x+y),e^{y-x}\tanh(y)\right)$$ Find the second-order Taylor expansion of $f$ about (x,y)=(0,0)$. I know how to find the Taylor ...
0
votes
2answers
33 views

How to put derivative of composition in Jacobian matrix?

Here are two functions: $f\left(u,v\right)=u^{2}+3v^{2}$ $g\left(x,y\right)=\begin{pmatrix} e^{x}\cos y \\ e^{x}\sin y \end{pmatrix} $ I need to make Jacobian matrix of $f\circ g$. I found ...
-1
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0answers
31 views

What if curl of vector field has more than 1 dimension?

Assume $v(x,y,z) = (xy, -2yz, z^2 - yz)$, then this field's curl is $(2y-z, 0, -x)$. So what am I to make of this, concerning the plane where the rotation occurs? It is known that if a field's curl ...
1
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1answer
33 views

Help needed understanding this explanation for the Jacobian

In this Quora Answer, a very intuitive explanation for the Jacobian is provided, there is however a step I don't understand: He takes this square: And via a polar coordinates transformation ...