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).

learn more… | top users | synonyms (1)

1
vote
1answer
262 views

Finding the partial differential of given problem

Given: $$u=ax+bx^2+u^4$$ Find the partial derivative with respect to $x$ and $y$ and as well total derivative of $u$. Please tell me complete description of this question.
5
votes
5answers
564 views

total differential (thermodynamic relations)

Assume that: there are four quantities $S,T,P,V$; any two of them can be varied independently, but the other two are then determined ---(1) $dU=TdS-PdV$ ---(2) $F=U-ST$ ---(3) then, according to ...
4
votes
2answers
76 views

Finding the most general class of solutions to $x\partial_{y}f = y\partial_{x}f$

Consider the following PDE for $f(x, y)$: $$ x\frac{\partial f}{\partial y} = y\frac{\partial f}{\partial x}\tag{1} $$ Clearly, one can separate the variables, so take $f(x, y) = p(x)q(y)$: $$ ...
2
votes
6answers
87 views

Why $\vec{r}$ is commonly use for vector equation?

I'm wondering why $\vec{r}$ is commonly use in mathematics (vector calculus, line integrals) and physics for denote the vector equation. Edit/Added clarification: I'm wondering why the letter $r$ is ...
3
votes
2answers
141 views

Optimization for constrained problem

I'm reading about Lagrange multipliers from a Pattern recognition book appendix and on one point the following is stated: $\begin{align} ...
1
vote
0answers
49 views

Why is there no analytic antiderivative of a multivariate Gaussian?

I've been working through (with some help) the standard formulae for multivariate Gaussians. But I can't find anything on analytic formulae (even with error functions) for the antiderivatives. Why ...
0
votes
1answer
46 views

Rotation matrix for non-isometry transformation

Imagine that you have a sphere in $\mathbb{R}^3$ and a plane (that is parallel to the x,y plane) through the sphere. Now you want to have a clockwise rotation in the x/y plane that does the ...
2
votes
1answer
51 views

Differentiability in $\mathbb{R}^n$: how to prove that $f'(a)v=g'(a)v$ for all $v \in \mathbb{R}^m$?

Let $f,g:U\to\mathbb{R}^n$ be differentiable at point $a\in U$, where $U\subset\mathbb{R}^m$ is an open set. Suppose $f(a)=g(a)$. Prove that $$\lim_{v\to0}\frac{f(a+v)-g(a+v)}{|v|}=0\;\;\;\;[\#]$$ if, ...
4
votes
1answer
155 views

Why does $\frac{\partial^2f}{\partial x \partial y} = \frac{\partial^2f}{\partial y \partial x}$

I'm studying multivariable calculus and I came across the following property of partial derivatives: $$\frac{\partial^2f}{\partial x \partial y} = \frac{\partial^2f}{\partial y \partial x}$$ where ...
2
votes
2answers
250 views

Average value using triple integrals

Find the average value of the function $f(x,y,z)=xyz$ over the tetrahedron with vertices, $(0,0,0),(1,0,0), (1,1,0), \text{and }(1,1,1)$
0
votes
1answer
45 views

Finding the limits in the order dydxdz

I need help finding the limits to the triple integral: $$\iiint\limits_{E}{f(x,y,z)}\: dy\, dx\, dz$$ where $E$ is bounded by the plane, $2x+y=2$ and the cylinder, $z=y^2$. I'm also having trouble ...
3
votes
0answers
75 views

Taylor Polynomial in Multivariable Case

I am doing Problem 9.30 in Rudin's Principles of Mathematical Analysis and have done part (a) and (b) but got stuck on part(c). In part (b) I have finished the proof that: $$f(\mathbf a+\mathbf ...
1
vote
1answer
59 views

det function in concave

Let $f(A)=(\det(A))^{\frac{1}{n}}$. And assume domain of $f$ is space of positive semi definite symmetric $n\times n$ matrices with real entries. Show that $f$ is concave: $$f((1-t)A+tB)) \ge ...
2
votes
1answer
71 views

surfaces $F$ and $G$ are tangent if and only if $\nabla{F}\times{\nabla{G}}=\mathbf{0}$

Suppose that two surfaces are given by the equations $F(x,y,z)=c$ and $G(x,y,z)=k$. Moreover, suppose that these surfaces intersect at the point $(x_0,y_0,z_0)$. Show that the surfaces are tangent ...
1
vote
1answer
110 views

Source for limits at infinity in higher dimensions?

Suppose $f:\mathbb{R}^2\to\mathbb{R}$ and we define things like $f(x,+\infty):=\lim_{y\to+\infty}f(x,y)$. Then how would one define, say, $f(+\infty,-\infty)$? Typically for limits not at infinity ...
3
votes
2answers
116 views

what path reduces magnetic field strength

I'm having trouble solving this particular problem in Colley's Vector Calculus. I believe my trouble lies in not being able to set up a differential equation. Here is the problem; Igor, the ...
1
vote
1answer
36 views

How can I determine which numbers of a set total to a given sum.

In Excel, I have a list of 30 different numbers ranging from $100$ to $1000$. If I have a sum representing several of those variable numbers, is there a way for me to determine which possible ...
1
vote
1answer
90 views

Showing that the function$f(x,y)=x+y-ye^x$ is non-negative in the region $x+y≤1,x≥0,y≥0.$

ok, since it's been so long when I took Calculus, I just wanna make sure I'm not doing anything wrong here. Given $f:\mathbb{R}^2\rightarrow \mathbb{R}$ defined as $f(x,y)=x+y-ye^x$. I would like to ...
-4
votes
3answers
190 views

Derivative of $(x^TAx)^2$

Let $x \in \mathbb{R}^N$, $A \in \mathbb{R}^{N \times N}$ What is the derivative of $(x^TAx)^2$ with respect to x?
1
vote
1answer
131 views

Evaluating part of a triple integral by changing from rectangular coordinates to cylindrical coordinates

Seeing that this is my first time posting, I hope I'm following the rules correctly. Anyways, the question I'm stuck on is: Evaluate: $$ \int_{-3}^3\int_{-2}^2\int_{-\sqrt{9-y^2}}^\sqrt{9+y^2} ...
5
votes
0answers
99 views

the spectrum and determinant of the Laplacian on $S^3$

I came across the following statement in a paper: On $S^3$, the eigenvalues of the vector Laplacian on divergenceless vector fields is $(\ell + 1)^2$ with degeneracy $2\ell(\ell+2)$ with $\ell ...
3
votes
2answers
287 views

What is the difference between differential topology and calculus on manifolds?

I'm trying to teach myself one and bought a book on the other. It seems to me that they both cover about the same material. This leads to the question: What is the difference between differential ...
2
votes
1answer
1k views

Divergence of stress tensor in momentum transfer equation

Let suppose that we work in a 2D cartesian coordinates. what will be x and y components of $\nabla.\left[-p I+\mu \left(\nabla \text{u}+(\nabla \text{u})^T\right)-\frac{2}{3} \mu (\nabla.\text{u}) I ...
0
votes
2answers
1k views

the partial derivative of $f(x,y)=\ln(x+\sqrt{x^2+y^2}), f_x (3,4)$

I know the answer is $\dfrac{1}{8}(1 +\dfrac{3}{5} )$ but i do not understand where the 1 inside the bracket comes from, I know how to get the $\dfrac{1}{8}$ and $\dfrac{3}{5}$ can someone clarify?
1
vote
2answers
97 views

Does this hold for three numbers [duplicate]

If $a\ge b\ge c\ge0$, does it hold that $\sqrt[3]{\left(a-b+c\right)^{2}}\ge\sqrt[3]{a^{2}}-\sqrt[3]{b^{2}}+\sqrt[3]{c^{2}}$? Thanks for any help.
2
votes
1answer
63 views

help with a problem on implicit fn theorem

Let $f: {\mathbb R}^{n+k} \rightarrow {\mathbb R}^n$ be a $C^1$ map. Suppose that $f(a)=0$ and $Df(a)$ has rank n. Show that if $c$ is a point in ${\mathbb R}^n$ sufficiently close to $0$, then the ...
2
votes
1answer
163 views

Question about integral equations

Consider the equation $$g(t) = \int_a^b K(t,s)f(s) ds $$ where $g$ and the kernel $K$ are known and $f$ is to be determined. Suppose that the equation has a solution. Under what conditions on the ...
3
votes
2answers
146 views

Double integral with messy variable substitution

I was looking through old multivariable analysis exams and found this double integral (with solution). My problem is I can't seem to understand how the transformation to the new area of integration is ...
3
votes
0answers
60 views

partial derivative notation question

I'm reading a book called Correlated Data Analysis, Analytics, and Applications and I simply don't understand some notation. The author says, in chapter 2, page 26: A unit deviance is called ...
1
vote
2answers
121 views

Laplace operator's interpretation (Laplacian)

What is your interpretation of Laplace operator? When evaluating Laplacian of some scalar field at a given point one can get a value. What does this value tell us about the field or it's behaviour in ...
4
votes
1answer
86 views

If $a\ge b\ge-c\ge0$, is $\sqrt[3]{a-b-c}\ge\sqrt[3]{a}-\sqrt[3]{b}-\sqrt[3]{c}$?

Let $a\ge b\ge-c\ge0$. Is it true that $\sqrt[3]{a-b-c}\ge\sqrt[3]{a}-\sqrt[3]{b}-\sqrt[3]{c}$?
2
votes
1answer
46 views

Why in this proof we get $\alpha \geq 0$?

I've solved the following problem: "Let $u,v \in \mathbb{R}^n$ with $u \neq 0$ be such that $|u+v|=|u|+|v|$ (euclidean norm), show that there's $\alpha \in \mathbb{R}$ with $\alpha \geq 0$ such that ...
3
votes
2answers
46 views

For $a > 0$, $x \in \mathbb{R}^n$, show that $x \mapsto \frac{ax}{\sqrt{a^2 - |x|^2}}$ is smooth

Let $a > 0$, and set $B_a = \{x \in \mathbb{R}^n : |x|^2 < a \}$. Let $\phi : B_a \to \mathbb{R}^n$ be given by $\phi(x) = \frac{ax}{\sqrt{a^2 - |x|^2}}$. Prove that $\phi$ is a diffeomorphism ...
6
votes
1answer
51 views

Is this proof that the vectors are colinear correct?

I was solving the following exercise: "Let $x,y \in \mathbb{R}^n$ be nonzero such that if $z$ is orthogonal to $x$ then $z$ is orthogonal to $y$. Prove that $x$ and $y$ are colinear". My idea was: ...
1
vote
3answers
130 views

How to find the derivative of $F(t) = \int_0^t f(t, x) \, dx$?

Let $f: \mathbb{R}^2 \to \mathbb{R}$ be a continuous function. Define $F: \mathbb{R} \to \mathbb{R}$ by, $$ F(t) = \int_0^t f(t, x) \, dx $$ Then, I'm not sure how to get $F'$. If, there are functions ...
1
vote
2answers
107 views

Does $\frac{\partial \dot{r}}{\partial \dot{q}}=\frac{\partial r}{\partial q}?$

Would it considered abuse of notation to say something in the grounds of $$\frac{\partial \dot{r_i}}{\partial \dot{q_j}}=\frac{\partial r_i}{\partial q_j}?$$ It is supposed to follow from ...
1
vote
1answer
537 views

Given an airplane's latitude, longitude, altitude, course, dive angle and speed, how do you find the new latitude, longitude, and altitude?

I have a model for an airplane that includes the aforementioned information. I want to be able to watch this model fly around on a map over time. Finding distance traveled is easy (distance = speed * ...
1
vote
0answers
158 views

Optimization of three variables with a constraint

I have a question puzzling me for a while. I tried using Lagrange multipliers, however it began to get messy as well as the fact that i am new to the method of Lagrange multipliers! The question is ...
1
vote
1answer
72 views

Mathematical question concerning Lagrange multipliers of a Lagrangian

In Lagrangian Mechanics I have in general holonomic constraints of the form $f(q_1,...,q_n,t)=0$ and then I am able to use the method of Lagrange multipliers, where I go from a Lagrangian $L$ to a ...
1
vote
1answer
95 views

product rule for partial derivative conversion

Working problems in Colley's Vector Calculus and I'm refreshing on partial derivatives, in particular the product rule and chain rules. I've come across a problem where we're identifying cartesian ...
2
votes
2answers
131 views

Exam question: Multivariable calculus, differentiation

I've decided to finish my education through completing my last exam (I've been working for 5 years). The exam is in multivariable calculus and I took the classes 6 years ago so I am very rusty. Will ...
1
vote
2answers
221 views

Trouble with gradient intuition

I'm currently learning about gradients, and I thought khanacademy could help me acquiring some intuition. The actual computation is clear to me, however I'm having trouble understand the intuition. ...
3
votes
0answers
213 views

Partial derivative equality

I'm working through Colley's Vector Calculus 3rd edition to reacclimate and introduce myself to all things vectors and differential forms. The following question is stumping me. Suppose $z=f(x,y)$ ...
4
votes
2answers
82 views

What is a directional derivative?

I have encountered this in an online PDE course I'm following but I've never really been exposed to it. I've looked for the 'formal' definitions but I've never really understood any concept by looking ...
1
vote
2answers
36 views

Show that $f:I\to\mathbb{R}^{n^2}$ defined by $f(t)=X(t)^k$ is differentiable

Let $I$ be a interval, $\mathbb{R}^{n^2}$ be the set of all $n\times n$ matrices and $X:I \to\mathbb{R}^{n^2}$ be a differentiable function. Given $k\in\mathbb{N}$, define $f:I\to\mathbb{R}^{n^2}$ by ...
2
votes
1answer
120 views

Integration formula with wedge product.

On $\mathbb{R}^3$, consider a compactly supported $2$-form $$\omega = f_1 \, dx_2 \wedge dx_3 + f_2 \, dx_3 \wedge dx_1 + f_3 \, dx_1 \wedge dx_2.$$ Choose for $S$ the parametrization $h: ...
1
vote
1answer
67 views

Calculate: $\int_A\mbox{ div }v\, d(x,y,z)$

$A:=\left\{(x,y,z)\in\mathbb{R}^3:x^2+y^2\leq 1,0\leq z\leq 1\right\},$ $v\colon\mathbb{R}^3\to\mathbb{R}^3, (x,y,z)\longmapsto (x^3,x^2y,zx^2)$. Calculate $\int_A\mbox{div }v\, ...
0
votes
1answer
91 views

Double Integral: $f(x,y)=x$ if $x=y$, and $f(x,y)=0$ otherwise

Let $\Omega = [0,1] \times [0,1]$. Let $f \colon \Omega \to \mathbb{R}$ be $f(x,y)=x$ if $x=y$, and $f(x,y)=0$ otherwise. I would like to show the integral exists or not using the criterion of Riemann ...
6
votes
0answers
132 views

1-form with positive integral over a path

Given any smooth path $\gamma$ on a manifold $M$, when can we construct a 1-form $\omega \in \Omega^1(M)$ so that $\int_{\gamma} \omega > 0$? It is easy to see that such $\omega$ exists if the path ...
1
vote
0answers
49 views

Calculating $\int_A\langle v,n\rangle\, dS$

Consider $$ A:=\left\{(x,y,z)\in\mathbb{R}^3: x^2+y^2\leq 1, 0\leq z\leq 1\right\},\\ v\colon\mathbb{R}^3\to\mathbb{R}^3, (x,yz)\longmapsto (x^3,x^2y,zx^2). $$ Calculate $$ ...