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
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2answers
85 views

Application of Green's Theorem when undefined at origin

Problem: Let $P={-y \over x^2+y^2}$ and $Q={x \over x^2+y^2}$ for $(x,y)\ne(0,0)$. Show that $\oint_{\partial \Omega}(Pdx + Qdy)=2\pi$ if $\Omega$ is any open set containing $(0,0)$ and with a ...
0
votes
2answers
36 views

If $σ$ is an exact differential $1$-form on the plane, then the form $ω=σ+xdy$ is not exact

If $σ$ is an exact differential $1$-form on the plane, then prove that the form $ω=σ+xdy$ is not exact. In the previous part of the question we have calculated the integral of the differential ...
1
vote
0answers
38 views

Really confused by multivariable calculus physical interpretations.

I have been learning MV calculus and have become really confused by what these things actually mean, for example: $$\iint f(x,y)~dA$$ should be calculating the volume under a surface in 3 dimensions ...
2
votes
1answer
40 views

Notation for integral of a vector function over an ellipsoid

For a short proof, I need to write a point $\pmb y\in\mathbb{R}^p$ as the integral of the surface of the ellipse $\pmb x^{\top}\pmb Q\pmb x=c$ where $\pmb Q$ is a $p$ by $p$ PSD matrix (for now ...
3
votes
3answers
166 views

Why spherical coordinates is not a covering?

Maybe this is an idiot question and I'm committing a trivial mistake. Let $\phi (\theta, \varphi) = (\cos \theta \sin \varphi, \sin \theta\sin \varphi, \cos \varphi)$ be the usual covering of the ...
0
votes
1answer
116 views

Identify the surface whose equation is given.

I have a test tomorrow and need help with the following question: Identify the surface whose equation is given: $p=3\sec(\varphi)$ So I know that I can multiply both sides by p to get ...
4
votes
0answers
57 views

Exercise from Pugh's Real Analysis Regarding Zero Derivative on Open Subsets of $\mathbb{R}^m$

Assume that $U$ is a connected open subset of $\mathbb{R}^n$ and $f:U\to\mathbb{R}^m$ is differentiable everywhere on $U$. If $(Df)_p=0$ for all $p\in U$, show that $f$ is constant. I immediately ...
8
votes
5answers
594 views

Determine whether or not the limit exists: $\lim_{(x,y)\to(0,0)}\frac{(x+y)^2}{x^2+y^2}$

Determine whether or not the limit $$\lim_{(x,y)\to(0,0)}\frac{(x+y)^2}{x^2+y^2}$$ exists. If it does, then calculate its value. My attempt: $$\begin{align}\lim \frac{(x+y)^2}{x^2+y^2} &= ...
1
vote
1answer
21 views

Image of Diffeomorphism to $R^n$

If a diffeomorphism $F$ is defined by $F:V\rightarrow \Bbb R^n$, does this mean $F(V)=\Bbb R^n$ ? My textbook keeps explicitly using $F(V)$ but I don't understand how this could be anything other than ...
1
vote
0answers
26 views

Double integral over carved circular disc [Answered in comments]

Compute $$\iint_D \ln(1+x^ 2+y^2) \,dx\,dy, \, D= \{(x,y); 1 \leq x^2+y^2 \leq 2\} $$ Looks like standard integrals to me. I change to polar coordinates and we get $$\iint_E r\ln(1+r^2) \, ...
0
votes
1answer
123 views

Solving a multivariate non-linear system of equations using Newton's method

I'm doing a report on the mathematics of GPS, and I have the following equations to solve for $x$, $y$, $z$ and $t_b$: I'm using Newton's method, with the Jacobian matrix of partial derivatives ...
3
votes
1answer
69 views

Mean Value Theorem Like Statement About Manifolds

Let $S$ be a connected $m$-dimensional embedded subamnifold in $\mathbf R^m\times \mathbf R^n$. Suppose that $S$ intersects $\{\mathbf 0\}\times \mathbf R^n$ at two different points. Conjecture: ...
2
votes
0answers
117 views

Find the derivative of f at point P in the direction of vector u.

Find the derivative of $f$ at point $(18,9)$ in direction of $\left<7,2\right>$. $$f(x,y) = \arctan \left(\frac{2y}{x} \right) + 3\arcsin\left( \frac{xy}{324} \right)$$ For this I got ...
1
vote
1answer
84 views

Showing that the gradient is orthogonal to level surface

It is well known that the gradient of a function (which is sufficiently well behaving) $g(x)$ is orthogonal to its level surface, for example $g(x)=0$. I have seen the following derivation of this ...
0
votes
1answer
35 views

Work done by F (vector field) on C (curve)

Let $C$ be the rectangle, with vertices at points $(0,0)$, $(0,1)$, $(2,0)$, $(2,1)$ and an anticlockwise orientation. Let $$F=(P(x,y),Q(x,y))$$ be a vector field with $$P=y^2+e^{x^2}+ye^{xy}$$ and ...
0
votes
1answer
521 views

Find directions in which f increases and decreases the most rapidly. Then find the derivatives of f at these directions. [closed]

$f(x,y)=3x^2+2xy+4y^2$ This is what I have to find: direction of fastest increase I found the gradient vector: $f(x,y)=(6x+2y)i+(2x+8y)j$ $f(9,2) = \langle 58,34\rangle$ Is this the direction of ...
3
votes
2answers
86 views

Using spherical coordinates to find volume of a region

Use spherical coordinates to find the volume of the region lying above $z = \sqrt{3x^2+3y^2}$ and within the $x^2+y^2+z^2=2az$, $a>0$. So far I know that the first graph is a cone and the second ...
3
votes
1answer
49 views

Differentiation under integral sign (arctan-function)

I have the integral $$ F(s) = \int_{0}^{\infty} \frac{\arctan(sx)}{x(1+x^2)} dx$$ and am supposed to solve it by finding $F'(s)$. So we get $$ F'(s) = \int_{0}^{\infty} \frac{\partial F}{\partial s} ...
0
votes
1answer
61 views

Question about double summation notation.

Just started learning about double integrals literally $10$ minutes ago. I have a fairly good grip on the Riemann integral and so far it seems very similar, but we are just working with volumes ...
4
votes
1answer
45 views

Trying to prove that a function got no limit at $(0,0)$

Let $f:\mathbb{R}^2\rightarrow \mathbb{R}$, defined by: $$ f(x,y)=\begin{cases} 1 & y=x^{2}\\ 0 & \text{otherwise} \end{cases} $$ How can I show that this function got no limit at $(0,0)$? ...
1
vote
1answer
56 views

Implications of zero divergence ($\nabla \cdot F$) when finding the flux

Say we are given a vector field $$F=(-x^2/2+xy,xy+y^2,-3yz-3)$$ with the property $\nabla\cdot F=0$. If we would like to find the flux through the part of the surface $x^2+y^2+2z^2=3$ that lies ...
1
vote
0answers
95 views

Exercise about max and min of a 2D function with absolute value

I haven't done an exercise like this so, please, tell me if the proceeding is wrong and any kind of observations that you think can help me. Find global max and min of $$f(x,y)=|x^2-y|$$ in ...
4
votes
1answer
77 views

What are higher derivatives?

From Wikipedia: Higher derivatives can also be defined for functions of several variables, studied in multivariable calculus. In this case, instead of repeatedly applying the derivative, one ...
3
votes
1answer
112 views

Extreme values of a two-variable polynomial

Is it possible to find a two-variable polynomial which has only two extreme values on the whole plane, one is a local maximum, another is a local minimum, and the local maximum is less than the local ...
2
votes
2answers
109 views

Finding extreme values of a variable on an intersection of a sphere and a plane

Determine the minimum and maximum value of the variable $z$ defined by the curve given by: \begin{cases} x^2+y^2+z^2=1 \\ x+2y+2z=0 \end{cases} So do I need to find a function $z=f(x,y)$ or just ...
1
vote
2answers
59 views

Need visualization advice for learning partial derivatives and calculus with more than one variable.

Okay so I just recently started learning calculus with more than one variable and whilst I'm coming to grips with many of the ideas and stuff I'm finding it difficult to visualize certain things for ...
1
vote
0answers
88 views

Taylor series of a twice-differentiable scalar function

I've come across this passage somewhere on wikipedia: If $f(t,x)$ is a twice-differentiable scalar function, its expansion in a Taylor series is $$df = \dfrac{\partial f}{\partial t}dt + ...
1
vote
1answer
51 views

Differential calculation in multiple variables function (cannot understand 2nd order differential form)

This question is somehow related to this question. Consider a multiple variables function $G(u, v) = \left(\matrix{x(u,v) &=& G_x(u,v)\\y(u,v) &=& G_y(u,v)\\z(u,v) &=& ...
3
votes
1answer
56 views

Multivariable Calculus with Tensors

I'm looking for a book at the undergraduate level on multivariable calculus (for a 2nd course of multivariable calculus) that introduces and makes use of tensors to describe higher order derivatives ...
2
votes
0answers
41 views

Finding area of a spheroid

Let $M=\{(x,y,z)\in \Bbb{R}^3 : (x/a)^2 + (y/b)^2 + (z/c)^2 = 1\}$. Find $\text{vol}_2(M) = \int_M 1 dS$. My attempt: The map $$\Phi:(0,\pi)\times (0,2\pi)\to \Bbb{R}^3\\ \qquad (\varphi, ...
7
votes
3answers
536 views

Why is this map called a fold?

Consider the map $\varphi : \mathbb R^2 \to \mathbb R^2$ defined by $(x,y) \mapsto (x,y^2)$. Apparently this map is called a fold as the $(x,y)$-plane is folded over and creased along the axis ...
0
votes
1answer
19 views

Derivative of the magnitude of a parametric function

I am trying to show that $d/dt$ $|r(t)|^2 = r(t)*r'(t)$, where $r(t)= <x(t), y(t), z(t)>$ and $r(t) \neq 0$. I first tried using the fact that $|r(t)|^2 = (x(t))^2+(y(t))^2+(z(t))^2$ and then ...
1
vote
2answers
57 views

Remembering the definition of the Jacobian: any tips?

I find it impossible to remember that the Jacobian of $f: \mathbb R^n \to \mathbb R^m$ is $$ \begin{pmatrix} {\partial f_1 \over \partial x_1} & {\partial f_1 \over \partial x_2} & \dots ...
0
votes
1answer
45 views

What does $\|u\|_{\mathcal{C}^2(\bar{\Omega})}$ mean?

What might $$\|u\|_{\mathcal{C}^k(\bar{\Omega})}$$ mean? $u$ is a sufficiently often differentiable function $\Omega \rightarrow \mathbb{R}$ and $\Omega \subset \mathbb{R}^n$ a bounded domain. It ...
0
votes
0answers
11 views

Does the gradient gives a natural orientation in a manifold? [duplicate]

I want to solve the following problem: Let $A\subset\mathbb{R}^n$ open $g:A\to\mathbb{R}$ of class $C^{1}$ and $g'(x)\not=0$ in each $x\in A$ then I want to compute $dV$ in the differentiable ...
0
votes
0answers
60 views

How to check whether the following function is concave or convex or neither.?

Let $\pi$ be a vector such that all its elements sum to 1. i.e, $\sum_1^n \pi(i) = 1$ where $\pi(i)$ denotes the i$^{th}$ component and $n$ is the length of the vector. Let $D$ be a diagonal matrix ...
1
vote
1answer
49 views

Converting an integrand into a polylog?

Compute the integral $$\int_0^1 dx\,dy\, \frac{\ln(1+y(1-x))}{1-xy}$$ I was just wondering if there is a way to convert the integrand into a polylog? This comes from a tutorial following a lecture ...
0
votes
1answer
24 views

Is the following property suffictient for second order differentialbility?

Let $U\subset R^n$ be an open set, and $f:U\to\mathbb R$ a $C^1$ function. Suppose that for any $x_0\in U$, there exists a $n$-variable-polynomial $T_{x_0}$ of degree at most $2$ satisfying that ...
1
vote
3answers
1k views

Derivation of divergence in spherical coordinates from the divergence theorem

I'm trying to find the expression of the divergence of a vector field $\vec{E}$ in spherical coordinates from the theorem : $$\iint_{S(V)}(\vec{E}.\vec{n})dS = \iiint_{V}div(\vec{E})dV$$ but if I ...
5
votes
1answer
77 views

The integral of a closed form along a closed curve is proportional to its winding number

Source: Guillemin-Pollack Exercise 4.8.2. Let $\gamma$ be a smooth closed curve in $\mathbb{R}^2 - \{0\}$ and $\omega$ any closed $1$-form on $\mathbb{R}^2 - \{0\}$. Prove that$$\oint_\gamma ...
4
votes
0answers
89 views

Matrix Calculus

My school offers Matrix Calculus class next semester. I have never heard about this subject before and got intrigued. After a short chat with professor I found myself unable to get rid of suspicion ...
3
votes
1answer
56 views

Does map induced by rotation preserve the volume form?

Let $A: \mathbb{R}^n \to \mathbb{R}^n$ be a rotation. My question is, does the map of $S^{n-1}$ onto $S^{n-1}$ induced by $A$ necessarily preserve the volume form?
0
votes
0answers
18 views

Prove that unitary normal vector to a manifold is $\nabla g / |\nabla g|$ [duplicate]

I want to solve the following exercise: Let $A\subset\mathbb{R}^n$ open $g:A\to\mathbb{R}$ of class $C^{1}$ and $g'(x)\not=0$ in each $x\in A$ then I want to compute $dV$ in the differentiable ...
0
votes
2answers
307 views

Determine if z is a function of x and y. $6x-4y+2z=10$

"Determine if z is a function of x and y. $6x-4y+2z=10$. Find the formula" All i did was equate for z $$z = 5-3x+2y$$ That is the formula. And It's pretty obvious that the answers are unique but i ...
0
votes
0answers
48 views

How do you find the inverse of a multivariable function?

In 1D variable calculus, you have a nice theorem that says: Suppose $f$ is differentiable and has an inverse on $I$. Suppose $x_o \in I$ and $f'(x_0) \neq 0$. Let $y_o = f(x_o)$, then ...
8
votes
1answer
68 views

Angle form, 1-form, proof verification.

Check that the $1$-form $d\,\text{arg}$ in $\mathbb{R}^2 - \{0\}$ is just the form$${{-y}\over{x^2 + y^2}}\,dx + {{x}\over{x^2 + y^2}}\,dy.$$ My solution is as follows. Observe that we can ...
0
votes
0answers
23 views

How to go about drawing slices and projections of iterated integrals

I have no idea how to go about drawing/graphing slices parallel to an iterated integral. Specifically this one: Triple Integral: $$\large{\int_0^1 \int_y^1 \int_y^x xe^{z^2} dzdxdy}$$
2
votes
1answer
225 views

Application Stokes's Theorem

I am a bit unsure the way Stoke's theorem is applied in this case. Evaluate $\oint\limits_C {xydx + yzdy + zxdz} $ around the triangle with vertices $(1,0,0), (0,1,0), and (0,0,1)$, oriented ...
0
votes
0answers
19 views

Drawing slices and projections of an iterated integral.

I'm having a rough time visualizing and graphing the slices and projections of this iterated integral: $\int \limits _0 ^1 \int \limits _y ^1 \int \limits _y ^x x \mathbb e ^{z^2} \space \mathbb d x ...
2
votes
2answers
55 views

Find minimum value of multivariable-function

A tent with 2 rectangle shaped sides (no floor) and 2 isosceles triangles shaped gables with the volume $V$ is to be constructed. Determine the height so that the minimum amount of cloth is needed. ...