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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5
votes
1answer
134 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 ...
5
votes
5answers
222 views

Interpreting higher order differentials

I'm trying to understand Taylor's Theorem for functions of $n$ variables, but all this higher dimensionality is causing me trouble. One of my problems is understanding the higher order differentials. ...
5
votes
2answers
700 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
645 views

if the curvature is constant and positive, then it is on the circunference

I'm trying to prove that if $\alpha(t)=(x(t),y(t))$ is a $C^2$ regular curve $(\alpha'\neq0)$ with constant and positive curvature, then $\alpha$ is on the circunference and if $\alpha$ is the ...
3
votes
3answers
99 views

Changing order of integration (multiple integral)

Prove $$ \int_0^a\left( \int_0^x \left( \int_0^y \left( \int_0^z f(u) \, du \right) dz \right) dy \right) dx = \int_0^a \frac {(a-t)^3}{3!} f(t) dt $$ where $a$ is constant. So I began with two ...
3
votes
2answers
190 views

Find maximum of $P$

Let $$P = \frac{{{x^2}}}{{{x^2} + yz + x + 1}} + \frac{{y + z}}{{x + y + z + 1}} - \frac{{1 + yz}}{9}.$$ Find maximum of $P$ where $x, y,z$ are nonnegative real numbers such that ${x^2} + {y^2} + ...
3
votes
1answer
894 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.
2
votes
4answers
710 views

Example of a function $f:\mathbb{R}^2\to\mathbb{R}$ not differentiable at $(0,0)$, but has a directional derivative at $(0,0)$ in all directions

Give an example of a function $f:\mathbb{R}^2\to\mathbb{R}$ such that $f'_u(0,0)$ exists in all directions $\|u\| = 1$, but $f$ is not differentiable at $(0,0)$. You have to show that your example ...
2
votes
2answers
244 views

Using nabla with partial derivatives and the Laplace operation $\partial_x^2+\partial_y^2+\partial_z^2$

Source of the problem p.812 here. Suppose $$\bar{F}(x,y,z)=(xy-z^2)\bar{i}+(xyz)\bar{j}+(x-y^2-z^2)\bar{k}.$$ I am concerned where I need to nabla an unit vector for example with $$\triangledown ...
1
vote
1answer
96 views

Calculate surface area of a F using the surface integral

Task Given: $$F := \{(x,y,z) \in \mathbb{R}^3 \mid (x,y) \in W,z=f(x,y)\}$$ Calculate the surface area using the surface integral: $i) \; f(x,y) := x+y \;\; and \;\; W := [12,31] \times ...
11
votes
5answers
341 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 ...
11
votes
3answers
350 views

Determine whether $\lim_{(x,y)\to (2,-2)} \dfrac{\sin(x+y)}{x+y}$ exists.

I am trying to determine whether $\lim_{(x,y)\to (2,-2)} \dfrac{\sin(x+y)}{x+y}$ exists. I should be able to use the following definition for a limit of a function of two variables: Let $f$ be a ...
11
votes
1answer
628 views

Problem 3-38 in Spivak´s Calculus on Manifolds

This is not homework. Problem 3-38 reads: Let $A_{n}$ be a closed set contained in $(n,n+1)$. Suppose that $f:\mathbb{R}\rightarrow \mathbb{R}$ satisfies $\int_{A_{n}}f=(-1)^{n}/n$ and $f(x)=0$ for ...
8
votes
1answer
114 views

Continuity of normalized displacement vector for a smooth closed curve

I am currently working on chapter 3.12 of "Differential Equations and Dynamical Systems" by Lawrence Perko. I am stuck on the continuity of the function $g$ in Theorem 3. My work (up to the prove of ...
7
votes
1answer
328 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
494 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
67 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
95 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 ...
5
votes
1answer
3k views

Linear transformation applied to a multivariate Gaussian random variable - what is the mean vector and covariance matrix of the new variable?

Given a random vector $\mathbf x \sim N(\mathbf{\bar x}, \mathbf{C_x})$ with normal distribution. $\mathbf{\bar x}$ is the mean value vector and $\mathbf{C_x}$ is the covariance matrix of ...
4
votes
2answers
499 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 $T=\{x=f(u,v); y=g(u,v)\}$ ...
4
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 ...
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
979 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
190 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
2k 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
61 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
2k views

Help evaluating triple integral over tetrahedron

I have a triple integral of $\iiint xyz\,dx\,dy\,dz$ 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 ...
2
votes
2answers
260 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
223 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
101 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
135 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
212 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
261 views

Proving that $\nabla \times (U(r) \hat{r} = 0 $

I was just checking to see if I wsa doing this right, as it isn't a formal proof. Just showing the identity. Let $U(r) \hat{r}$ b a vector in spherical coordinates. Given that the vector is only ...
2
votes
3answers
1k views

How to interpret Hessian of a function

I know that gradient of a function gives the direction in which the directional derivative of the function is maximum. Is there any similar interpretation of Hessian ?
2
votes
2answers
141 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
41 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
1answer
80 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
200 views

Minimum area of the parallelepiped surface

Among all the retangular parallelpipeds of volume $V$, find one whose total surface área is minimum Using the Lagrange Multipliers method, I've found that it is a cube with dimensions $ \sqrt[3]{V} ...
1
vote
2answers
83 views

Use implicit function theorem to show $O(n)$ is a manifold

In class today our teacher mentioned that one can use the implicit function theorem to show that $O(n) \subseteq \mathbb{R}^{n^2}$ is a submanifold...that is, map $A \mapsto A^* A$, and set it equal ...
1
vote
1answer
165 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
483 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 ...
1
vote
1answer
203 views

Chain rule for multivariable functions confusion

Suppose $f=f(x,y(x))$. Then applying the chain rule we get $\frac{\partial f}{\partial x}=\frac{\partial f}{\partial x}\frac{\partial x}{\partial x}+\frac{\partial f}{\partial y}\frac{\partial ...
0
votes
2answers
81 views

Evaluating a triple integral

$$\iiint \sqrt{x^2+z^2}\,dx\,dy\,dz$$ I am asked to evaluate this integral over the region $$D:=\left \{ (x,y,z) \in\mathbb{R}^3 :x^2+y^2+z^2 \leq 1 \wedge x^2+y^2+z^2 \leq 2y\right \}$$ I did manage ...
0
votes
2answers
86 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 ...
9
votes
5answers
502 views

Are derivatives defined at boundaries?

Given a differentiable function $f : [-5,5] \rightarrow \mathbb{R},$ I was under the impression that the derivative $f'$ has domain $(-5,5).$ However, according to Wikipedia ...a differentiable ...
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 ...