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
198 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
1answer
147 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
129 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
38 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
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
51 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
142 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
398 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
54 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 ...
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5answers
290 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 ...
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
203 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 ...
5
votes
1answer
1k 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 ...
4
votes
4answers
161 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
2answers
153 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
411 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
1answer
77 views

Without Stokes's Theorem - Calculate $\iint_S \operatorname{curl} \mathbf{F} \cdot\; d\mathbf{S}$ for $\mathbf{F} = (-y^3,x^3,z^3)$ - 2012 9C

Question: 2012 9C. Consider the (cutoff) paraboloid defined by $z= x^2 + y^2 , \frac{1}{9} \le z \le 1$. Sketch the surface. Verify Stokes’s Theorem for for $\mathbf{F} = (-y^3,x^3,z^3)$. Herein, I ...
3
votes
1answer
598 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
2answers
162 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
218 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
118 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
89 views

How to partial differentiate a total differential and be rigorous on all the notion?

Start with $$dS=\left(\frac{\partial S}{\partial T}\right)_VdT+\left(\frac{\partial S}{\partial V}\right)_TdV$$ Using the notes shown here Method 1: i) Divide both sides by dV ...
2
votes
1answer
88 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
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
2answers
83 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
137 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
182 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
240 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 ...
2
votes
1answer
364 views

Find the Jacobian

Find the Jacobian $$\frac{\partial(x,y)}{\partial(u,v)}$$ for $x=u^2+v^2$, $y=u^2-v^2$. My solution: I tried solving it as it is by using the Jacobian matrix (determinant?) and got my answer to be ...
2
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?
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vote
3answers
36 views

When is this vector valued function pointing towards the origin?

"A fighter plane, which can shoot a laser beam straight ahead, travels along the path $\mathbf{r}(t) = \langle 5 - t, 21 - t^2, 3 -\frac{1}{27}t^3\rangle$. Show that there is precisely one ...
1
vote
2answers
68 views

Double Integration with change of variables

I am having trouble with the following double integral: $$\iint\limits_D(x^2+y^2) \;dA$$ where $D$ is given by the region enclosed by the curves $xy=1$ $xy=2$ $x^2-y^2 =1$ $x^2-y^2 =2$ I have ...
1
vote
1answer
140 views

Express partial derivatives of second order (and the Laplacian) in polar coordinates

$z=f(x,y)$ where $x=rcosθ$ and $y=rsinθ$ Find $ \frac{\partial z}{\partial x}$ and $ \frac{\partial^2 z}{\partial x^2}$ I'm having big troubles with using chain rule, in particularly the second ...
1
vote
1answer
77 views

Continuity conditions for multivariate functions.

Is the following true ? A proof or counter-example or reference would be nice. A function $f:\mathbb{R}^2 \rightarrow \mathbb{R}$ is continous at $(0,0)$ if and only if if for all $a, b$, the limits ...
1
vote
1answer
67 views

Solve Multivariate Polynomial

Is there a general way to solve a multivariate polynomial (example here: http://mathworld.wolfram.com/MultivariatePolynomial.html) Say for instance I knew some function $F(x,y) = xy + x^2 + y^3 + ...
1
vote
1answer
2k views

Vector equation and parametric equation for a line segment

Suppose that $P = (1,1,7)$ and $Q = (8,6,1)$. Inside parenthesis are x-coordinate value, y-coordinate and z-coordinate. The question is to find vector and parametric equation for a line segment. Now I ...
1
vote
1answer
276 views

shortest distance between two points on $S^2$

Length of Curve in $2D$ is $l_{\gamma}(\mathbb{R}^2)=\int_{0}^{1}\sqrt{(dr/dt)^2+r^2(d\theta/dt)^2}$ Length of a curve in $3D$ is ...
1
vote
1answer
80 views

Question Concerning Vectors

I am given the information that $ \vec{u} = \langle 1, 1/2 \rangle$ and $\vec{v} = \langle 2,3 \rangle$. There are a few pieces I am asked to find, and these are the one I am having trouble with: ...
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vote
2answers
209 views

Do we need additional assumptions for problem 3-37 (b) in Spivak´s calculus on Manifolds?

Problem 3-37 (b) reads: Let $A_{n}=[1-1/2^{n},1-1/2^{n+1}]$. Suppose that $f:(0,1)\rightarrow \mathbb{R}$ satisfies $\int_{A_{n}}f=(-1)^{n}/n$ and $f(x)=0$ for any $x\notin$ any $A_{n}$. Show that ...
1
vote
1answer
172 views

Hölder continuity of a function from $[0,1]$ to $[0,1]^2$

I'm trying to prove that if $g: [0,1] \longrightarrow [0,1]^2$ is an $\alpha$-Hölder continuous mapping whose image is the entire square $[0, 1]^2$ then $\alpha \leq 1/2$. I wouldn't know where ...
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vote
0answers
150 views

Help to show this equality of differential operators in spherical coordinates

There are two points on the same sphere with coordinates ${R, \theta_1, \phi_1}$ and ${R, \theta_2, \phi_2}$. Also I have the operator $\displaystyle { \nabla _{{\Omega _1}}^2 + \nabla _{{\Omega ...
1
vote
1answer
223 views

passing the Derivative inside an integral

Question: Suppose we have: $F(x)=\int_{a(x)}^{b(x)}e^{h(x,t)}dt$. Is it true that $F^{'}(x)=\int_{a(x)}^{b(x)}\frac{\partial h(x,t)}{\partial x}.e^{h(x,t)}dt$ ? Please tell me under what ...
0
votes
1answer
39 views

Differentiating a vector valued function

If I have a function $y(x)=f(a+x(b-a))$ where $a, b$ are constant vectors, and $y: \mathbb{R} \rightarrow \mathbb{R}$, what would $\frac{dy}{dx}$ be in terms of $f$? I know the chain rule would be ...
0
votes
1answer
23 views

line integral (multivariable calculus)

Evaluate the line integral $$ \int_C (\ln y) e^{-x} \,dx - \dfrac{e^{-x}}{y}\,dy + z\,dz $$ where C is the curve parametrized by $r(t)=(t-1)i+e^{t^4}j+(t^2+1)k$ for $0\leq t\leq 1$
0
votes
2answers
42 views

Second Order Partial Differentiation

I don't have a clue on how to start this question. I have a feeling I will need to use the Clairaut's theorem: $f_xy=f_yx$ Can anyone advise?
0
votes
2answers
104 views

2 examples to try to understand partials derivatives and deriviability

To prove that a functions has partial derivatives every partial has to exist, and every partial exist only if the limit of definition of partial exist. Is this right? Then if partials exist ,and the ...
0
votes
0answers
52 views

Fréchet normal cone

Given $x\in \Omega(\subset X)$ (X: Banach space) and $\varepsilon\geq 0$, the set of $\varepsilon-$normals to $\Omega$ at $x$ by \begin{align} \widehat N_\varepsilon(x;\Omega):=\left\{x^*\in X^*\mid ...
0
votes
1answer
79 views

Triple Integral Volume Question

The question asks for the triple integral of $e^{-(x^2+y^2+z^2)^{3/2}} dV$ where $D$ is a sphere of radius $4$. The answer that I came up with is $2(1-e^{-64})$. However, I am not confident in this ...