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
3answers
161 views

Derivative of function with 2 variables

I've leart in Calculus 1 that the derivetive of $f(x)$ is: $$\lim_{h\to0} \frac{f(x+h) - f(x)}{h}$$. suppose $f(x,y)$ is a function with 2 variables, does $$f'(x,y) = \lim_{h\to0} \frac{f(x+h, y+h) ...
26
votes
4answers
2k views

Volume of Region in 5D Space

I need to find the volume of the region defined by $$\begin{align*} a^2+b^2+c^2+d^2&\leq1,\\ a^2+b^2+c^2+e^2&\leq1,\\ a^2+b^2+d^2+e^2&\leq1,\\ a^2+c^2+d^2+e^2&\leq1 &\text{ ...
0
votes
1answer
52 views

Given that $|f(x,y)| \le x^{2}y^{2}$, prove that $f$ is differentiable at (0, 0).

Given that $f : \mathbb{R}^{2}\rightarrow \mathbb{R}$ is a function such that $|f(x,y)| \le x^{2}y^{2}$ for all $(x,y) ∈ R^{2}$, prove that $f$ is differentiable at $(0, 0)$. I know that I should ...
2
votes
2answers
213 views

partial derivative chain rules

Suppose that there is $f(a,b)$. Also suppose that $b = g(a, \text{and some other variables})$. By chain rule, it seems that $$\frac{\partial f}{\partial a} = \frac{\partial f}{\partial ...
2
votes
1answer
60 views

How to find the partial derivative of this function?

Lets say I have a function:$$\nu=\frac{RT}{P}+B_{p}(T)RT$$ and I am trying to find $\left(\frac{\partial \nu}{\partial T}\right)_{P}$. I understanding that the partial derivative of the first term is ...
3
votes
0answers
154 views

Spivak Calculus on Manifolds Exercise 2-9

I'm kind of stumped on this exercise in two spots. First I'll state the problem: Two functions $f,g : \mathbb{R} \to \mathbb{R}$ are equal up to $n$th order at $a$ if $$\lim_{h \to 0} \frac{f(a + ...
4
votes
4answers
146 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.
2
votes
0answers
81 views

Verifying the Divergence Theorem for Half of a Sphere

Here is an exercise that I was assigned for homework: .......................................................... To the bottom left, I have scanned an example problem for verifying the divergence ...
0
votes
1answer
29 views

Multivariable Calc Proof [duplicate]

Suppose that $F(x,y,z)=0$ is an equation so that any variable can be solved in terms of the other two. Show that $\displaystyle \frac{\partial x}{\partial y}\cdot\frac{\partial y}{\partial z}\cdot ...
1
vote
1answer
106 views

The shortest path connecting three points

I have 3 points X,Y,Z, lets call them buildings. I need to find the shortest amount of path that connects the 3 buildings, these buildings can be in any sort of shape and any distance from each ...
0
votes
1answer
40 views

Finding the volume of two subsets of $\mathbb{R}^3$

Define two subsets of $\mathbb{R}^3$. $$A = \{ (x,y,z) \in \mathbb{R}^3 : |x|+ |y| + |z| \leq 1 \} $$ $$B= \{ (x,y,z) \in \mathbb{R}^3 : \max{ \{ |x|,|y|,|z| \} } \leq 1 \}$$ Find vol($B$)-vol($A$) ...
0
votes
2answers
16 views

Graph the region of a two variable inequation

I need help with $( x + y -1) ( x - y +1) y <= 0$ There is a known method to conclude what area of the XY plane satisfy that inequation?. I usually have the same problem with other inequations ...
0
votes
0answers
28 views

Show that $f: K_1(0) \rightarrow \mathbb{R}^3$ is Lipschitz.

Firstly, the Assignment: Let $V = (\mathbb{R}^3 ,\|\cdot\|_{\infty})$ where $\|\cdot\|_{\infty}$ denotes the maximum norm and consider the function: $$f: K_1(0) \rightarrow ...
0
votes
1answer
88 views

Volume in toroidal coordinates

I'm asked to determine the volume of a 3d object described with toroidal coordinates. As we never treated this kind of coordinate system before, I'm given the following transformation: $x = ...
0
votes
1answer
56 views

Differentiation Formula for Moving Regions.

I've run into a few calculations in a series of textbooks/papers that require differentiating an integral with a changing region. In particular, I'd like to know if $f(x,t):\mathbb{R}^d\times ...
4
votes
1answer
136 views

Product of Elements in SU(2)

Let $$ V := \frac{x_4+i\vec{x}\cdot{\vec{\sigma}}}{\left|x\right|}$$ where $\left(x_1,x_2,x_3,x_4\right)\in\mathbb{R}^4$, $|x|$ is the Euclidean norm, and $\sigma^j$ are the Pauli matrices. Let ...
3
votes
1answer
82 views

gradient of norm square of a random vector

Let $g(w)= \|Y_n - f(w,X_n) \|^2$ where $f:\Bbb R^d \times \Bbb R^m \to \Bbb R^k : w \in \Bbb R^d$. What is the gradient of $g$ ? $X_n$ and $Y_n$ are random vectors. Basically, I want to find ...
0
votes
1answer
37 views

A notational confusion on gradient

Given a parametrized function $f_{w}: \Bbb R ^{m} \to \Bbb R ^{k}, w \in \Bbb R^d$, I see in a book the following notation $\bigtriangledown ^ {w} f_{w}(.)$ denote its gradient w.r.t. $w$. What is the ...
1
vote
0answers
41 views

When the euclidean distance criteria does not work?

Good day to everyone. I have a question about classification in presence of additive gaussian noise. Assuming a predefined reference set of $N$ complex-valued vectors ...
2
votes
1answer
780 views

Iterated Integral requiring u-substitution

This is an iterated integral. I have tried solving it several times using u-substition, but I am not getting the correct answer. My latest result is (4/15)(10^(5/2)-33). Obviously something is off, ...
1
vote
1answer
48 views

Constructing a function from level sets

Suppose we know what the projection of the level sets into the xy-plane of some function $f:\mathbb{R}^2 \rightarrow \mathbb{R}$ looks like. How can I construct a closed form for $f$ by "lifting" the ...
1
vote
2answers
31 views

Area of $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$

Could you tell me how to calculate the area of part of the plane: $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$, $a, b, c >0$ where all coordinates of a point are positive?
2
votes
3answers
106 views

How can ${\iint\limits_{D}{{e^{x^2+y^2}}}}dxdy$ be found?

How can ${\iint\limits_{D}{{e^{x^2+y^2}}}}dxdy $ be found, if $D$ is $x$ O $y$ axis? So far I have done it this far: ...
0
votes
0answers
47 views

Integral of multivariate normal density function

Is anybody know a suited close-form solution for this integral: $$ I=\int_{R^n} x_i \cdot x_j \cdot f_N({\bf x},{\bf \mu},{\bf \Sigma}) d{\bf x} $$ where ${\bf x}=\{x_1,\ldots,x_n\}$ and $f_N$ is the ...
1
vote
1answer
70 views

Maximum distance from the origin to the surface

I am having trouble getting the maximum distance from the origin to the surface $$ \frac{x^4}{16} +\frac{y^4}{81} + z^4 = 1 $$ Knowing I have to maximize $x^2 +y^2+ z^2$ and that the constrain ...
0
votes
2answers
51 views

Limit of $\frac{\sin(xy^2)}{x^2+y^2}$ as $x\to0$ and $y\to0$

Does anybody have an idea, how to solve this limit, I cant figure it out. I wanted to try some algebraic magic to get something with $\frac{\sin(x)}{x}$ but without success. Any sugestions? Thank you. ...
1
vote
2answers
36 views

A question on a multivariable continuously differentiable function

Assume $f(x_{1},x_{2})$ is a real-valued continuously differentiable function, and assume it holds that $x_2D_{1}f(x_1,x_2) - x_1D_2f(x_1,x_2) = 0$ where $D_1$ is the partial derivative with respect ...
4
votes
1answer
99 views

Explanation of differential forms and notation

I'm doing multivariable calculus and I'd love if someone could shed some light on things that confuse me. When we did integrals of real functions with real variables, the $dx$ that was in every ...
3
votes
3answers
82 views

Unique solution for parametric system of two equations

The system is: $$x - 2y + z = 2\alpha \\ 3(xy + xz + yz) = 3\alpha - 4$$ I have to find for which values of $\alpha$, the system has an unique solution. I've tried to simplify both expressions, ...
1
vote
0answers
57 views

How to calculate double limit in Maxima?

I want to calculate a double limit in Maxima. For instance,$$\lim_{(x, y)\to(0, 0)}\frac{x^3y}{x^6+y^2}$$ By the way, it is very interesting that I calculated it in "wolframalpha" yet it gave me the ...
1
vote
1answer
409 views

Finding the shortest distance between two planes using Lagrange multipliers

A problem (among a list of Lagrange multipliers problems in Earl Swokowski's Calculus) states as follows: find the shortest distance between $2x+3y-z = 2$ and $2x+3y-z=4$. I can see that the ...
1
vote
2answers
78 views

Divergence Proof

I'm not sure where to start on this proof...also, my book didn't give any clarification as to what $f$ and $\textbf{F}$ are. Usually $F$ is a scalar function, and $\textbf{f}$ is the vector field ...
1
vote
0answers
13 views

Line integral using centroid and area

Let $C$ be a simple closed curve in $\mathbf{R^2}$ with positive orientation enclosing a region $D$. Assume that D has area equal to 2 and centroid $(\bar{x}, \bar{y})=(3,4)$. Let $\mathbf{F}= \langle ...
1
vote
1answer
46 views

Cannot understand question

Can anyone please help me understand the meaning of this question? So let $f(x,y) = x^2y - 3xy^3 = z$ $P = (1,1,1)$ Equation of tangent plane at $P$: $z = -x-8y+7$ Next the question says 'The unit ...
0
votes
2answers
29 views

Multivariable Differentials Problems

I am having a lot of trouble with this question: For $j=1,...,n$, define the function $f_{j}$ on $R^{n}$\ {0} by $f_{j}(x)=x_{j}$/|x|. Show that $\sum_{i=1}^n x_{j}df_{j}$=0.
1
vote
0answers
37 views

Gradient Proof Question

Prove the given formula ($r=||{\textbf{r}}||$ is the length of the position vector field $\textbf{r}(x,y,z)=x\textbf{i}+y\textbf{j}+z\textbf{k}$). $$\nabla \dfrac{1}{r} = \dfrac{-\textbf{r}}{r^3}$$ ...
1
vote
0answers
60 views

Divergence and Curl in Spherical Coordinates

Using these definitions, how would you solve for div $\textbf{f}$ and and curl $\textbf{f}$?What are $f_p$, $f_\theta$, and $f_\phi$? Thanks.
4
votes
2answers
71 views

Which Cross Product for the Desired Orientation of a Hyperboloid ? [Stewart P1103 16.9.8]

P1103 16.9.$8.$ Evaluate the surface integral $\iint_S \mathbf{F} \cdot d\mathbf{S}$. $\mathbf{F} = (x^3y,-x^2y^2,-x^2yz)$ and $S$ is the surface of the solid bounded by the hyperboloid $x^2 + ...
2
votes
1answer
70 views

Direction of gradient from level surface?

In the diagram below, we see a level surface with a gradient. As a consequence of the multivariable chain rule, the gradient is normal to the surface. That's clear to me. Why is the gradient ...
1
vote
0answers
229 views

Continuity of a piecewise defined function in two variables

I need some insight into the “approach” that I used to solve this problem. Namely, I was asked to find if the following function is continuous on all $\mathbb{R}^2$: $$ f(x, y) = \left\{ ...
1
vote
1answer
33 views

local invertibility does not imply global invertibility

What is an example of a smooth function with continuous derivatives, that is locally invertible but not globally, and the reason for that is not injectivity. My first idea was $f:\mathbb{R}^{2}\to ...
0
votes
1answer
29 views

Help With Understanding a Specific Derivative

I'm working through the MIT 18.02 2007 Multivariate Calc class right now, and I don't understand a derivative they use in their answer key. Specifically, given $T = \frac{-a\sin tI +a\cos tJ + ...
0
votes
0answers
50 views

Exercise on Jacobians/Triple Integrals

Q: Integrate the function $f=x^2+y^2$ over an elliptical cone, with the base being the ellipse $\frac{x^2}{4}+y^2=1$, $z=0$ and the apex at the point $(0,0,5)$. The integral can be made simpler by ...
2
votes
3answers
83 views

Convexity of $\sqrt{x^2+y^2}$

I am to prove that $\sqrt{x^2+y^2}$ is convex for $x,y>0$. Intuitively, if I look at the derivatives, $\frac{x}{\sqrt{x^2+y^2}}$, $\frac{y}{\sqrt{x^2+y^2}}$, they are increasing in every positive ...
5
votes
1answer
174 views

curl$(\mathbf{F} \times \mathbf{G})$ with Einstein Summation Notation [Stewart P1107 16 Review.20]

I'm aware of another post on this vector identity, but I have a question on the derivation based on P47 of Source: ...
3
votes
0answers
39 views

Brouwer's fixed theorem using Stokes' theorem

according to Wikipedia, there is a simple way to prove Brouwer's fixed point theorem from Stokes' theorem: see here. So I would like to present the former famous theorem (Brouwer's one) to my Calculus ...
2
votes
1answer
76 views

curve integral - intersection between plane and sphere

I am going to calculate the line integral $$ \int_\gamma z^4dx+x^2dy+y^8dz,$$ where $\gamma$ is the intersection of the plane $y+z=1$ with the sphere $x^2+y^2+z^2=1$, $x \geq 0$, with the orientation ...
0
votes
2answers
92 views

Implicit function theorem problem

I have the function $$(x-2)^3y+xe^{y-1}=0$$ And I have to see if $y$ can be described as a function of $x$ around (1,1). The implicit function theorem can't be applied in this case. What should I ...
1
vote
1answer
43 views

Double integral inequality

Proof this inequality: $$\int_1^4 \int_0^1 (x^2+\sqrt{y})\cos(x^2y^2) dx dy\leq 9 $$ I don't know how to approach to this, any idea ?
3
votes
0answers
33 views

A singular $n-$cube and a circumference defined the border than 2-cube

This is an exercise from "Calculus on Manifolds" by Michel Spivack (first edition, p.100): If $c$ is a singular $1$-cube in $\mathbb{R}^2-\{0\}$, with $c(0)=c(1)$, show that there is an integer ...