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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3
votes
2answers
67 views

$\int_{x=0}^{100}\int_{y=0}^{100-x} \int_{z=0}^{100-x-y}(x+y+z)^{10} \mathrm dx \, \mathrm dy \, \mathrm dz$

Is there any easy way to calculate the following $\int_{x=0}^{100}\int_{y=0}^{100-x} \int_{z=0}^{100-x-y}(x+y+z)^{10} \mathrm dz \, \mathrm dy \, \mathrm dx$ and ...
3
votes
2answers
180 views

Curvature of a regular parametrization

Prove that if $\mu: [a,b] \to \mathbb{R}^n$ is a regular parametrization of a curve then the curvature at $\mu(t)$ is given by: $$\kappa(t) = ...
0
votes
1answer
64 views

Point of tangency for a circle between two vector

I'm having two vectors p and q starting at point O (origin). These vectors are known, as well as the origin point is. I know the angle α (alpha). Given a circle with arbitrary radius r, I want to be ...
1
vote
1answer
37 views

Solving surface integral

I just need some help on solving surface integral. Actually I already finished doing part (a) and part (b) but just part (c) I dont know how to do it. It would be nice if someone able to guide me to ...
5
votes
3answers
96 views

Is there another method?

If $x$ and $y$ are positive numbers such that $x + y = 1$, find the maximum value of $x^4y + xy^4$. I could do this problem my simplifying the expression to $xy(1-3xy)$ and taking $k=xy$, forming ...
6
votes
1answer
152 views

Problem 4-25 from Spivak's Calculus on Manifolds

I am reading through Spivak's Calculus on Manifolds and have come across a technicality in one of the problems that is annoying me. It is Problem 4-25, the statement of which is Let $c$ be a ...
1
vote
2answers
108 views

Integral in spherical coordinates, $\Omega$ is the unit sphere, of $\iiint_\Omega 1/(2+z)^2dx\ dy\ dz$

$$\iiint_\Omega \frac{1}{(2+z)^2}dx\ dy\ dz$$ There is a VERY similar question How to integrate $\iiint\limits_\Omega \frac{1}{(1+z)^2} \, dx \, dy \, dz$ here But this is different. I like my ...
8
votes
3answers
740 views

Proof for the funky trace derivative : $d (\operatorname{trace} (ABA'C))$?

Given three matrices $A$, $B$ and $C$ such that $ABA^T C$ is a square matrix, the derivative of the trace with respect to $A$ is: $$ \nabla_A \operatorname{trace}( ABA^{T}C ) = CAB + C^T AB^T $$ ...
3
votes
2answers
216 views

derivative of exponential of matrix trace

What is the derivative of $\sum_{ij}e^{-d_{ij}^2(X)}=\sum_{ij}e^{-\operatorname{tr}(X^TC_{ij}X)}$, w.r.t $X$ where $C_{ij}$ is a constant matrix and $d_{ij}^2(X)$ denotes the squared Euclidean ...
0
votes
1answer
153 views

Verifying Stokes Theorem on Vector Calculus

My question is to verify Stokes Theorem. I manage to do using Stokes Theorem $\iint_R \nabla\times\vec{F}\cdot d\textbf{S}$ and got my answer $7/6$ but I dont know how to do the direct line ...
1
vote
1answer
334 views

Direction of the resultant displacement Math help

Problem: A disoriented physics professor drives a distance 3.15km north, then a distance 2.50 km west, and then a distance 1.30km south. Find the direction of the resultant displacement, using the ...
1
vote
1answer
155 views

Direction and Magnitude of a Dog Running

Problem: A dog in an open field runs 12.0m east and then 30.0m in a direction 54 degrees west of north. Part A: In what direction must the dog then run to end up 12.0m south of her original starting ...
3
votes
1answer
140 views

Boltzmann's transformation and change of variables.

Boltzmann’s Transformation is used (among other things) to convert Fick's second law into an easily solvable ordinary differential equation. It uses the variable $\xi=\frac{x}{2 \sqrt{t}}$. As far ...
1
vote
1answer
66 views

Show that a partial differential eqution is satisfied (is my notation okay?)

Let: $f:\mathbb{R}\rightarrow\mathbb{R}$ be differentiable. $u:\mathbb{R}^2\rightarrow\mathbb{R}$ and $u(x,y)=e^{-y}f(x+y^2)$ Show that: ...
2
votes
4answers
2k views

Find the shortest distance between the point $(8,3,2)$ and the line through the points $(1,2,1)$ and $(0,4,0)$

"Find the shortest distance between the point $(8,3,2)$ and the line through the points $(1,2,1)$ and $(0,4,0)$" $$P = (1,2,1), Q = (0,4,0), A = (8,3,2)$$ $OP$ = vector to $P$ $$PQ_ = (0,4,0) - ...
1
vote
1answer
345 views

Finding the local extrema of this trigonometric, multivariate function

QUESTION Find all extrema and their places for $$ f(x,y) = \mathtt{sin} x + \mathtt{cos} y + \mathtt{cos} (x-y)$$ for $ 0 \le x \le \frac{\pi}{2}$ and $ 0 \le y \le \frac{\pi}{2}$ ATTEMPT I go ...
2
votes
1answer
79 views

How to prove that the circumference is $C^k$?

Notation: $U$ is a open subset of $\mathbb{R}^n$; $\partial U$ is a boundary of $U$; A $C^k$ function is a function $k$-times continuosly differentiable; $B(x_0,r)$ is a ball in $\mathbb{R}^n$ with ...
2
votes
3answers
80 views

Showing that the product of vector magnitudes is larger than their dot product

QUESTION Show that $$|\mathtt{u} \cdot \mathtt v| \le |\mathtt u||\mathtt v|$$ ATTEMPT Let $ \mathtt {u,v} \in \mathbb R^n$ such that $ \mathtt u = u_1 x_1 + u_2 x_2 + ... + u_n x_n, \mathtt v = ...
3
votes
3answers
121 views

Interesting question in analysis

I am trying to prove this : Consider $\Omega \subset R^n$ ( $n \geq 2$) a bounded and open set and $u $ a smooth function defined in $\overline{\Omega}$. Suppose that $u(y) = 0$ for $y \in \partial ...
3
votes
1answer
110 views

$f(x,y)=\sqrt{|xy|}$

$f(x,y)=\sqrt{|xy|}$ I need to calculate partial derivatives at $(0,0)$, and conclude whether it is differentiable there. $f_x(x,y)=\lim_{h\to 0}{f(x+h,y)-f(x,y)\over h}=0=f_y(x,y)$, so can I ...
0
votes
2answers
6k views

Find an equation for the line that is parallel to the plane $2x - 3y + 5z - 10 = 0$ and passes through the point (-1, 7, 4)

"Find an equation for the line that is parallel to the plane $2x - 3y + 5z - 10 = 0$ and passes through the point (-1, 7, 4)" I'm just learning this and am pretty confused on how to do this problem. ...
2
votes
1answer
41 views

Simplify vector equation

I know that $div E=0$ and I know what $ curl E$ is. Further, I know what the vector laplacian of $E$ is. Now I want to simplify $\nabla \times (\nabla \times f(x,y,z) E(x,y,z))$, where ...
3
votes
1answer
197 views

Derivative of scalar field

Let $f : \mathbb{R}^n \to \mathbb{R}$ be given by $f(x) = x^TA^TAx - \lambda( x^T x - 1)$, where $A$ is an $n \times n$ matrix and $\lambda$ is a scalar. My question is how to compute the derivative ...
1
vote
1answer
60 views

What is the curl of this function

I have a vector field $E(r,\theta,\phi)=E_re_r+E_{\theta}e_{\theta}+ E_{\phi}e_{\phi}$(the small e's are unit vectors) in spherical coordinates and I know what $\nabla \times E$ is. Now I multiply ...
2
votes
1answer
94 views

Study of Matrix Calculus

I need to study matrix calculus such as integration, differentiation, differentiation of functions of determinants and inverse matrices and then also other matrix based calculations such as ...
2
votes
1answer
88 views

Differentiation in multivariable analysis.

I read somewhere that if $F:U\in \Bbb{R}^n\to \Bbb{R}^m$ is differentiable at $X$ (where $X\in U$), then there exists an $m \times n$ matrix $A$ such that $$F(X+h)=F(X)+A\circ h+G(h)\|h\|$$ where ...
1
vote
1answer
43 views

Help to find a triple integral

I need help to find the integration $\int\int\int_D (x^2+y^2+z^2)^{3/2}dxdydz$ Where $D: 1\leq x^2+y^2+z^2\leq 4$; $ x,y,z\geq 0$.
3
votes
1answer
75 views

Show: $\int_{\mathbb{R}^n}\mbox{div }a(x)\, d^nx=0$

For a function $f\colon\mathbb{R}^n\to\mathbb{R}$ we write $f=o(r^{\alpha})$ if to any $\varepsilon>0$ there exists a $R>0$ so that $$ \lvert f(x)\rvert\leq\varepsilon\lVert ...
2
votes
1answer
331 views

How does the gradient of a function show greatest slope for a function $f(x,y,z)$?

$$\nabla f(x,y,z)=\frac{\partial f}{\partial x}\mathbf{i}+\frac{\partial f}{\partial y}\mathbf{j}+\frac{\partial f}{\partial z}\mathbf{k}$$ $\nabla$ is the gradient operator. $(\nabla f).\mathbf{r}$ ...
6
votes
1answer
130 views

Is there any relationship between Cauchy-Riemann equations and vector fields on manifolds?

Well, suppose we have $f : \mathbb{C} \to \mathbb{C}$ analytic, then if $f = u + iv$ the functions $u,v : \mathbb{C} \to \mathbb{R}$ satisfy the Cauchy-Riemann equations: $D_1u=D_2v$ and $D_2u=-D_1v$. ...
1
vote
1answer
220 views

Partial derivative of a function with respect to a vector

I have the following error term E: $$E = \frac{1}{c}\sum_{\substack{ i<j}} \frac{[d_{ij}^* - d_{ij}]^2}{d_{ij}^*}$$ where $$c = \sum_{\substack{ i<j}}d_{ij}^*$$ and $$d_{ij} = ...
0
votes
2answers
2k views

calculate surface area by using double integral

The question is: Find the surface area of the part of the cylinder $$y^2+z^2=2z$$ that is cut off by the cone $$x^2=y^2+z^2$$ (answer=$16$) This question is from a section that discuss the ...
0
votes
1answer
69 views

Creating a function from known data and variable relationships

I'm developing a game and I need to create a predictable function while most of the variables are not 100% under my control. I will explain the practical situation: You have two characters, trying to ...
2
votes
2answers
260 views

Finding local and global extrema even when the determinant of the Hessian zero

I am trying to solve the following problem: Let $f: \mathbb R^2\rightarrow\mathbb R$ be a function defined by $$ f(x,y) = x^{2n} + y^{2n} - nx^2 + 2nxy - ny^2, $$ where $n$ is a natural ...
2
votes
0answers
206 views

local parametrization of regular surface

I am doing excercises of Do Carmo's dg of curves and surfaces Chapter 2.2 and need some help with the following excercise: Show that the set $S=\{(x,y,z)\in R^3;z=x^2-y^2\}$ a regular surface and ...
1
vote
3answers
77 views

How to find the integral by changing the coordinates?

Let R be the region in the first quadrant where $$3 \geq y-x \geq 0$$ $$5 \geq xy \geq2$$ Compute $$\int_A (x^2-y^2)\,dx\,dy.$$ I tried to use $ u= y-x, v= xy$ as my change of coordinates, but then I ...
0
votes
1answer
47 views

what is the equivalent for $d' = 0$ for 3D surfaces to detect “pinches”?

with a given function in $R^2$ you can detect a "pinch" in the graph when the first derivative is zero and assume that the graph is "smooth" when the same first derivative is not zero: what is the ...
7
votes
1answer
337 views

Partition of Unity in Spivak's Calculus on Manifolds

I have a question about partitions of unity specifically in the book Calculus on Manifolds by Spivak. In case 1 for the proof of existence of partition of unity, why is there a need for the function ...
1
vote
1answer
63 views

Integrating $\iint \hat{n} \, dS $ over a closed surface?

One of the exercises in the book Div, Grad, Curl, and All That is to show that $$ \iint_S \hat{n} \hspace{1mm} dS = 0$$ for every closed surface $S$, using the divergence theorem. I know the theorem, ...
1
vote
2answers
101 views

Why does Stokes' Theorem allow any surface to be used when calculating a line integral.

I'm trying to understand Stokes' Theorem, what I don't get is how it allows you to pick any surface as long as the boundary is the same. Let's say that the vector field is increasing in strength ...
0
votes
4answers
3k views

Finding critical points of f(x,y)

Find the critical point of $$ f(x,y) = 3x^3 + 3y^3 + x^3y^3 $$ To do this, I know that I need to set $$f_y = 0, f_x = 0 $$ So $$f_x= 9x^2 + 3x^2y^3$$ $$f_y = 9y^2 + 3y^2x^3$$ Then you solve for x, ...
0
votes
0answers
68 views

A Question Related to Fubini's Theorem

I was wondering if there is a integrable function $f: A \times B \to \mathbb{R}$ (where $A, B \subset \mathbb{R}^n$ are rectangles) such that $$ \int\limits_{A \times B} f = ...
4
votes
1answer
216 views

A triple integral

I used spherical coordinates and couldn't solve this triple integral. $$ \iiint_S \left ( \left ( x-a \right )^{2}+\left (y-b \right )^{2}+ \left ( z-c \right )^{2}\right )^{-\frac{1}{2}}dx ~ dy ~ dz ...
0
votes
1answer
105 views

How to define $z$ with $x$ and $y$ using implicit function theorem?

Let $c$ be a nonzero constant. Consider the fuction $$f(x,y,z) =x^3yz+xyz^3=c.$$ How to define $z$ as a function of $x$ and $y$ ? How to use the implicit function theorem to solve this kind of ...
1
vote
2answers
82 views

Equation of first variation for a flow

I am trying to derive the equation of first variation for a flow for a vector field. Things I am told: $\mathbf{F}$ is a vector field of class $C^1$ with a flow $\phi$ of class $C^2$. From here I ...
1
vote
1answer
161 views

How to find the inverse function of $f: \mathbb R^2\to\mathbb R^2$?

How to find the inverse function of $f: \mathbb R^2\to\mathbb R^2$? for example, $u= x+2y, v=xe^y$, how to find the inverse?
0
votes
1answer
736 views

Local extreme value & saddle point: multi variable calculus

I am asked to find all local extreme values & saddle points of $$f(x,y) = 2x^2 + y^2 - xy - 7y + 8$$ $$f_x(x, y) = 4x-y, \qquad f_y(x,y) = 2y-x-7$$ $$f_x(x,y) = 0, \qquad y = 4x$$ $$f_y(x,y) ...
3
votes
2answers
90 views

Compute: $\int_{0}^{1}\int_{x}^{1} e^{x/y}$

Compute: $\int_{0}^{1}\int_{x}^{1} e^{x/y}$ I just don't know how to compute this integral. I tried u = x/y, but that didn't really lead me anywhere. It was suggested by fellows on a IRC that I graph ...
7
votes
1answer
132 views

A stupid question on Stokes' theorem

Suppose I want to do the simple integral $I=\int_{ R^2} \frac{\mathrm{d}r\wedge\mathrm{d}\phi}{(1+r)^2}$. Just evaluating the integral one quickly gets $\left. -2\pi\frac{1}{1+r} \right ...
0
votes
1answer
335 views

Find all extrema for $f(x,y) = 3xy$ subject to the constraint $4x^2 + 2y = 48$

Find all extrema for $f(x,y) = 3xy$ subject to the constraint $4x^2 + 2y =48$. I put it into the form of: $3xy - \lambda(4x^2 +2y -48) = F(x,y,\lambda)$ $3xy - 4x^2\lambda -2y\lambda + 48\lambda$ ...