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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7
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1answer
312 views

Helix in a helix

I am trying to work out a "helix in a helix" mathematically. Intuitively I think of this as a steel cable, which is made up of a number of smaller steel cables all bound together in spiral. If I ...
7
votes
1answer
3k views

Why does the formula for calculating a reflection vector work?

The formula for calculating a reflection vector is as follows: $$ R = V - 2N(V\cdot N) $$ Where V is the incident vector and N is the normal vector on the plane in question. Why does this formula ...
6
votes
2answers
437 views

A little help integrating this torus?

Let $\mathbf{F}\colon \mathbb{R}^3 \rightarrow \mathbb{R}^3$ be given by $$\mathbf{F}(x,y,z)=(x,y,z).$$ Evaluate $$\iint\limits_S \mathbf{F}\cdot dS$$ where $S$ is the surface of the torus ...
5
votes
1answer
51 views

What does shear mean?

As I understand it, the gradient of a vector field can be decomposed into parts that relate to the divergence, curl, and shear of the function. I understand what divergence and curl are (both ...
5
votes
1answer
94 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 ...
4
votes
1answer
121 views

Can the following triple integral be computed via elementary calculus methods?

Consider the following triple integral: $$\int_0^{2\pi}\int_0^1 \int_0^1 xy\sqrt{x^2 + y^2 -2xy\cos(\theta)} \, dx \, dy \, d\theta$$ A solution was provided to this integral by Jack D'Aurizio ...
4
votes
2answers
638 views

Is the boundary of the unit sphere in every normed vector space compact?

I wanted to ask whether the boundary of the unit sphere in every normed vector space is compact? I know that this is true for simple examples, but how is it in general?
4
votes
1answer
804 views

Continuity of one partial derivative implies differentiability

Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ be a function such that the partial derivatives with respect to $x$ and $y$ exist and one of them is continuous. Prove that $f$ is differentiable.
4
votes
1answer
1k views

Limits using epsilon delta definition $f(x,y)=xy$ for functions of two variables

Prove: using $\epsilon$-$\delta$ definition, the limit of both $f$ and $g$ as $(x,y)\to (0,0)$ is $0$. $f(x,y)=xy$ $g(x,y)=\frac{xy}{x^2 +y^2+1}$ Also, for Q2 can I convert $g(x,y)$ to ...
3
votes
5answers
789 views

What is the intuition behind the unit normal vector being the derivative of the unit tangent vector?

I've seen the math, but... It just doesn't make sense to me. How is the slope going to point perpendicular to the vector that is clearly a straight line not going in that direction?
3
votes
2answers
180 views

Proof of: If $x_0\in \mathbb R^n$ is a point of local minimum of $f$, then $\nabla f(x_0) = 0$.

Let $f: \mathbb R ^n\to\mathbb R$ be a differentiable function. If $x_0\in \mathbb R^n$ is a point of local minimum of $f$, then $\nabla f(x_0) = 0$. Where can I find a proof for this theorem? ...
3
votes
1answer
1k views

How to find the boundary curve of a surface, like the Möbius strip?

I feel like I am missing a key piece of intuition in trying to understand this. I have just recently started using Stoke's theorem and I struggle to see what the boundary curve of surfaces are. In ...
3
votes
1answer
1k views

Wire mass line integral

Wire is given with $y=\sqrt{25-x}$ and density is given with $ \delta(x,y)=15-y$. Mass should be calculated using line integral. This is my first assignment in this area and I need help with ...
3
votes
2answers
2k views

What is the vector form of Taylor's Theorem?

I checked most of the posts about Taylor expansion with scalar functions. Could anyone tell me what is the multivariate version of Taylor's Theorem, and how I can use it?
2
votes
0answers
54 views

Advanced Real Calculus - Differentiability

I struggling with some problems. Thank you for any help: This function is given : $ f(x,y)=(e^x-1)\frac y{(x^2+y^2)^\alpha}\;$ , and they ask the values of $\;\alpha\;$ for which f is can be defined ...
2
votes
2answers
237 views

A continuously differentiable function with vanishing determinant is non-injective?

(This question relates to my incomplete answer at http://math.stackexchange.com/a/892212/168832.) Is the following true (for all n)? "If $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ is continuously ...
2
votes
2answers
212 views

Problem using Stokes's Theorem - Boundary Curve, Unit Normal Vector [Stewart P1097 16.8.5]

$\Large{1.}$ How does one determine the boundary curve, denoted as C, to be the plane $z = -1$? I’m flummoxed because $S$ here is given as bottomless. I'm not enquiring about formal or rigorous ...
2
votes
3answers
98 views

Prove $\lim_{(x,y) \to (0,0)} \frac{x^3y}{x^4+y^2} = 0$

How would you prove the following limit? $$\lim_{(x,y) \to (0,0)} \frac{x^3y}{x^4+y^2} = 0$$ I think the best way is using the squeeze theorem but I can't find left expression. $$0 \le ...
2
votes
2answers
132 views

Multivariate differentiability verification

I have tried an attempt on the following: Let $F:U\in \Bbb{R}^n\to \Bbb{R}$ and $f:\Bbb{R}\to \Bbb{R}$ where $f$ is an even function. Now $F(\mathbf{x})=f(|\mathbf{x}|)$, where $|\ . |$ is the ...
2
votes
1answer
176 views

dropping a particle into a vector field

I'm independently studying Colley's Vector Calculus and am on the section on line integrals. I understand that the line integral gives the amount of work done on a vector field for a predetermined ...
2
votes
2answers
133 views

Maxima and minima of multivariable function $f(x,y)=6x^3y^2-x^4y^2-x^3y^3$

$$f(x,y)=6x^3y^2-x^4y^2-x^3y^3$$ $$\frac{\delta f}{\delta x}=18x^2y^2-4x^3y^2-3x^2y^3$$ $$\frac{\delta f}{\delta y}=12x^3y-2x^4y-3x^3y^2$$ Points, in which partial derivatives ar equal to 0 are: ...
1
vote
1answer
39 views

$k$-space tensor integral in statistical physics

$$Q=\int_{\text{all space}} \frac{\hbar \nu_g \mathbf{k}\mathbf{k}}{\exp[(\hbar \nu_g |\mathbf{k}|-\mathbf{k}\cdot\mathbf{u})/k_B T]-1}d\mathbf{k} $$ Please help me to integrate the above tensor ...
1
vote
2answers
1k views

Help evaluating triple integral over tetrahedron

I have a triple integral of dxdydz xyz over the volume of a tetrahedron with vertices (0,0,0), (1,0,0), (0,1,0), and (0,0,1). Normally I would just have limits 0 to 1 but that does not seem to work. ...
1
vote
1answer
78 views

Multivariable Calculus Order of Integration Question

I have a triple integral $ \int_{0}^{2} \int_{0}^{y^3} \int_{0}^{y^2} f(x,y,z) \ dz \ dx \ dy $. I have to find five different iterated integrals equivalent to this integral. I know that order of ...
1
vote
1answer
52 views

Compute $\iint\limits_R\frac{y}{x+y^2}dA$ where $R=[0,1]\times[1,2]$

Compute $\displaystyle\iint\limits_R\frac{y}{x+y^2}dA$ where $R=[0,1]\times[1,2]$ $\displaystyle\int_0^1\int_1^2\frac{y}{x+y^2}dydx=\int_0^1\int_1^2y(x+y^2)^{-1}dydx$ How do I integrate the ...
1
vote
1answer
150 views

Evaluate the line segment intergal

Evaluate the line integral $$\int_C xe^{y}\, {\rm d}s,$$ where $C$ is the line segment from $(-1,2)$ to $(1,1)$. I do not get this part of calculus at all please show me how this is solved and if ...
1
vote
4answers
449 views

Triangle integral with vertices

Evaluate $$I=\iint\limits_R \sin \left(\frac{x+y}{2}\right)\cos\left(\frac{x-y}{2}\right)\, dA,$$ where $R$ is the triangle with vertices $(0,0),(2,0)$ and $(1,1)$. Hint: use ...
0
votes
2answers
68 views

Green's theorem exercise

I am trying to solve the following problem: Show functions $P,Q:\mathbb R^2 \setminus \{(0,0)\} \to \mathbb R$ of class $C^1$ that verify $P_y=Q_x$ but $$\int_\gamma P(x,y)dy+Q(x,y)dy \neq 0$$ where ...
10
votes
5answers
313 views

Is line element mathematically rigorous?

I know differentials (in a way of standard analysis) are not very rigorous in mathematics, there are a lot of amazing answers here on the topic. But what about line element? $$ds^2 = dx^2 + dy^2 ...
6
votes
1answer
2k views

Is any divergence-free curl-free vector field necessarily constant?

I'm wondering, for no particular reason: are there differentiable vector-valued functions $\vec{f}(\vec{x})$ in three dimensions, other than the constant function $\vec{f}(\vec{x}) = \vec{C}$, that ...
5
votes
1answer
2k views

The composition of two convex functions is convex

Let $f$ be a convex function on a convex domain $\Omega$ and $g$ a convex non-decreasing function on $\mathbb{R}$. prove that the composition of $g(f)$ is convex on $\Omega$. Under what conditions is ...
5
votes
3answers
210 views

Equation of the Plane

I have been working through all the problems in my textbook and I have finally got to a difficult one. The problem ask Find the equation of the plane.The plane that passes through the points ...
4
votes
4answers
188 views

Convolution integral $\int_0^t \cos(t-s)\sin(s)\ ds$

How can I calculate the following integral? $$\int_0^t \cos(t-s)\sin(s)\ ds$$ I can't get the integral by any substitutions, maybe it is easy but I can't get it.
4
votes
3answers
3k views

Showing that a function is not differentiable

I want to show that $f(x,y) = \sqrt{|xy|}$ is not differentiable at $0$. So my idea is to show that $g(x,y) = |xy|$ is not differentiable, and then argue that if $f$ were differentiable, then so ...
4
votes
2answers
154 views

$\frac{d}{dx}(b^TAx)$ where $b, x \in R^{n\times 1}$ and $A \in R^{n\times n}$

How do you differentiate the following expressions with respect to the vector $x$. I think I might be a little conceptually confused on what happens when you take the derivative with respect to a ...
4
votes
2answers
455 views

Total Derivative and Multilinear Functions

So I'm starting to work through Spivak's Calculus on Manifolds and I'm having a little trouble verifying some of the claims made in the book problems. To review: Given a function ...
3
votes
2answers
349 views

Why absolute values of Jacobians in change of variables for multiple integrals but not single integrals?

If $g:[a,b]\to\mathbf R$ is a change of 1D coordinates, then the formula is: $$ \int_{g(a)}^{g(b)}\,f(x)\,dx = \int_a^b\,f(g(t))\frac{dx}{dt}\,dt. \qquad\text{(1)}$$ If $T=\{x=f(u,v); y=g(u,v)\}$ ...
3
votes
1answer
694 views

l'Hôpital's Rule and Multivariable Limits

I decided to post another message regarding this problem because I still didn't understand it at all: Can someone give me an example of function $f(x,y),g(x,y)$ for which: $\lim\limits_{r\to 0^+} ...
3
votes
1answer
228 views

Let $f(x,y)=\frac{x^3-y^3}{x^2+y^2}$. Is f differentiable in $(0,0)$?

Let $$f(x,y)=\frac{x^3-y^3}{x^2+y^2}$$ My solution manual says that this function is not diffb. in $(0,0)$ because it is not linear. Well my problem is that I don't see why this function is linear, ...
3
votes
1answer
119 views

Differentiability in $\mathbb{R}^{2}$

Here is my question: Find all the points $\left ( x,y \right )$ in $\mathbb{R}^{2}$ where the following function is differentiable: $f\left ( x,y \right )=\left | e^{x}-e^{y} \right |.\left ( x+y-2 ...
2
votes
1answer
92 views

Compute the flux of $(z \sin x, yz \cos x, x^2 + y^2)$ through the paraboloid.

Given the vector field $$F(x, y, z) = \langle z \sin x, yz \cos x, x^2 + y^2 \rangle,$$ calculate the flux $\int_S F \cdot \hat{n} \; dS$ through the paraboloid $$S = \{(x,y,z) : z = -3(x^2 + y^2) + ...
2
votes
1answer
131 views

Need help understanding a certain vector integral identity (Stokes' theorem corollary)

The Vector Integral page on the Wolfram mathworld website lists as eq.(4) the following vector integral identity: If $$\mathbf{F}:=\mathbf{c}\,F,$$ then ...
2
votes
0answers
59 views

Difficult multivarate random variable - how to calculate it?

I have a random variable defined by $Y=\frac{\sum_{j=1}^{N}l_j \cos\theta_j}{\sum_{j=1}^{N}l_j\sin\theta_j}$ where $l_j \sim \text{log-normal-distribution} (\left \langle l \right \rangle, \sigma _l)$ ...
2
votes
1answer
96 views

Finding $g_i:\mathbb{R}^n\to\mathbb R$ s.t $f(x)=\sum\limits_{i=1}^nx_i\cdot g_i(x)$

Let $f:\mathbb R^n\to\mathbb R$ differntiable and $f(0)=0$. Prove exist $g_i$ s.t for $x=(x_1,\dots,x_n):f(x)=\sum\limits_{i=1}^nx_i\cdot g_i(x)$. hint:$f(x)=\int\limits_0^1f\prime(tx)dt$. I dont ...
2
votes
2answers
87 views

How do I rotate a given point on $\mathbb{S}^n$ so that I send it to the South Pole

This seems pretty trivial but I'm not sure what to do. My coordinates are Cartesian, and I want to send point the $(x_1,...,x_{n-1},x_n)$ to point $(0,...,0,-1)$ so that all the other points are also ...
2
votes
1answer
142 views

Lagrange multiplier constrain critical point

When using Lagrange multipliers in an inequelity, $$ f(x,y) = x^2+y $$ with the constraint $$ x^2+y^2 \leq 1. $$ I have to find the critical points inside the "disk" right? I've done $$ f_x = 2x ...
2
votes
2answers
184 views

Other way to express $e^{|x|+|y|}$

I have a joint PDF with $e^{|x|+|y|}$. I know I can separate the function in two functions, $e^{|x|}$ and $e^{|y|}$. The limits for $x$ and $y$ are from $-\infty$ to $\infty$. Can I integrate from $0$ ...
2
votes
1answer
1k views

What is the meaning of evaluating the divergence at a _point_?

Reading this first, Divergence (div) is “flux density”—the amount of flux entering or leaving a point. Think of it as the rate of flux expansion (positive divergence) or flux contraction (negative ...
2
votes
1answer
245 views

Calculate the volume between $z=x^2+y^2$ and $z=2ax+2by$

I'm trying to calculate the volume between the surfaces $z=x^2+y^2$ and $z=2ax+2by$ where $a>0,b>0$. Here's what I've tried: First I noticed the projection of the volume to the xy plane is a ...
2
votes
2answers
1k views

A continuously differentiable map is locally Lipschitz

Let $f:\mathbb R^d \to \mathbb R^m$ be a map of class $C^1$. That is, $f$ is continuous and its derivative exists and is also continuous. Why is $f$ locally Lipschitz? Remark Such $f$ will not be ...