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
votes
1answer
12 views

Triple integration in cylindrical coordinates (obtaining the formula to integrate)

Can someone help me out with this? For some odd reason, I am able to derive the boundaries but cannot figure out the formula to use. I thought it was r^2 but apparently it's wrong.
0
votes
1answer
26 views

Geometric Interpretation of Liouville's Theorem?

The only bounded entire functions in $\mathbb{C}$ are constants. Could someone please give me a geometric interpretation of the theorem above?
1
vote
1answer
21 views

Differentiation of multivariate function with respect to another multivariate function [on hold]

How do I calculate the derivative $\frac{df(x,y)}{d(xy)}$, given that $x,y\neq 0$ and assuming that the derivative exists?
0
votes
1answer
15 views

Identify the Following Parametric Surfaces?

r(u,v) = ui+(ucosv)j+(usinv)k r(u,v) = ucos(v)i+usin(v)j+u^2k r(u,v) = ui+vj+(2u-3v)k r(u,v) = vi+cosvj+sinvk My Guess: Plane Circular Cylinder Cone Circular Parabloid Can someone please check ...
0
votes
1answer
15 views

w = x² - y² + 3z² direction with no change in w

Consider w = x² - y² + 3z². At (1, 1, 1), what is the fastest rate of change for w? What is a direction along which there is no change in w? I know how to do the first part, since the fastest rate of ...
1
vote
1answer
34 views

Show using the definition of limit $\lim _{(x,y)\to(0,1)}\frac{x^2-y^2}{x^2+y^2} = -1$

Show using the definition of limit that $$\lim _{(x,y)\to(0,1)}\frac{x^2-y^2}{x^2+y^2} = -1$$ and $$\lim_{ (x,y)\to(0,0)}\frac{ (1-\cos(xy))\sin y}{(x^2+y^2) }= 0$$ Definition of limit: ...
1
vote
2answers
28 views

Show a limit of a bounded function is 0 then solving the integral

$B=\{x^2+y^2\le1\}$, & for all $\delta>0$, there is $B_\delta=\{x^2+y^2\le\delta\}$. $f$ is a continuous function and $\|\nabla f\|\le1$ on $B$, and suppose $\frac{\partial^2f}{\partial ...
5
votes
5answers
68 views

Does $\left(\frac{\partial f}{\partial x}\right)^2=\frac{\partial^2f}{\partial x^2}$

This may be an obvious question but I'm just not thinking straight, thanks The answer must be no
1
vote
2answers
28 views

What do gradient, curl, and div input and output?

What do gradient, curl, and div input and output? (e.g. vector field or scalar function of several variables)
0
votes
2answers
15 views

line integral problem with a circle

I want to evaluate the line integral $$ \int_C (2xy^3+\cos x)\,dx + (3x^2y^2+5x)\,dy $$ where $C$ is the circle $x^2+y^2=64$ I parametrized the circle by $r(t)=8\cos t \ \hat{i} +8\sin t \ \hat{j}$ ...
0
votes
1answer
21 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 $0\leq t\leq 1$
1
vote
2answers
32 views

How to show that a line integral is independent of the path of integration

Show that the following line integral is path-independent $$ \int_C (\ln y) e^{-x} dx - \dfrac{e^{-x}}{y}dy + zdz $$
1
vote
1answer
15 views

evaluating line integral (multivariable calculus)

Evaluate the line integral $\int_C y(x^2+y^2)dx-x(x^2+y^2)dy+xydz$ where $C$ is parametrized by $r(t)=\cos t i+\sin t j+tk$ for $-\pi\leq t\leq \pi$. If I did it right (which I'm not sure if I did), ...
0
votes
3answers
39 views

Compute two-dimensional integral over a region bounded by circular arcs

How to compute $$ \iint_{M}y\,{\rm d}x\,{\rm d}y $$ where $ \ M\equiv\left\{\,% \left(\, x,y\,\right)\ \mid\ y\ \geq\ 0\,,\quad x^{2} + y^{2}\ \leq\ 1\,,\quad \left(\, x - 1\,\right)^{2} + y^{2}\ ...
0
votes
0answers
31 views

Partial derivatives of multivariable function with maximum

I have a question regarding the derivative of a maximum. I found the answer below which suggests evaluating the function at several intervals: Derivative of max function However, I have a function of ...
0
votes
0answers
16 views

Split this integral

I need to split this integral if possible: \begin{equation} \int_{\mathbb{R}^d} e^{\sum_{i=1}^dx_iz_i}cos(\sum_{i=1}^dy_iz_i)d\mathbf{z} \end{equation} I wanted split into two part : one with $x_i$ ...
1
vote
0answers
18 views

Level sets volume

Suppose that $f:\mathbb{R}^d\to\mathbb{R}$ is a nice function (whatever nice should mean), non-negative, with a compact support. Fix $v >0$ and define $$ A_{\epsilon} := \{x\in \mathbb{R}^d : v \le ...
1
vote
1answer
48 views

Calculus Partial derivative computation

I have the following systems of equation $$x^5v^2+2y^3u = 3 \\3yu - xuv^3 = 2$$ I need to show that this system defines v and u implicitly as functions of x and y near the point $(x,y,u,v) = ...
3
votes
1answer
49 views

Inner product, differential forms and surfaces (Stokes' theorem)

I'm trying to understand how do you get the Kelvin-Stokes theorem \begin{equation} \int_{S} (\nabla\times \omega) \cdot \mathrm{d}S = \int_{\partial S} \omega \cdot \mathrm{d}r \end{equation} from the ...
0
votes
0answers
14 views

Show that the function $f(x,y)=\int_b^yf_2(a,t)\ dt + \int_a^xf_1(t,y)\ dt $ is a potential function

Let $F=(f_1,f_2)$ be conservative over the open rectangle: $$R=\{(x,y):|x-a|<r,|y-b|<r\} $$ I need to show that the function $f(x,y)=\int_b^yf_2(a,t)\ dt + \int_a^xf_1(t,y)\ dt $ is a ...
1
vote
2answers
36 views

How to prove a function has no local minima.?

Suppose we have a function $ f:\mathbb{R}^2 \to \mathbb{R}$, of class $C^2$ that satisfies: $3\frac{\partial^2f}{\partial x^2}(x,y)+4\frac{\partial^2f}{\partial y^2}(x,y)=-1$, for all $(x,y) \in ...
0
votes
2answers
40 views

Why does this implicit differentiation formula fail?

Suppose we have that $$\frac{dy}{dx} = -\frac{y}{x}.$$ Taking the derivative implicitly with respect to $x$, we can easily obtain $$\frac{d^{2}y}{dx^{2}} = \frac{-\frac{dy}{dx}x + y}{x^{2}} = ...
1
vote
2answers
27 views

Do the partial derivatives of this piecewise constant function exist? If yes, how can I compute them?

Given this piecewise constant function $$ f(x,a,b,c,d,e) = \begin{cases} a, & x \lt d; \\ c, & d \le x \lt e; \\ b, & e \le x. \\ \end{cases} $$ do the partial derivatives ...
0
votes
0answers
43 views

Minimum of $x+y+z$ on $\{(x,y,z) \in \mathbb{R}^3 | z \le x^2+2y^2+3, z\ge 3x+2y\}$

Find the minimum of $x+y+z$ on $$\{(x,y,z) \in \mathbb{R}^3 | z \le x^2+2y^2+3, z\ge 3x+2y\}.$$ My first naive thoughts would be to consider setting up a nasty triple integral and evaluate it or ...
0
votes
0answers
17 views

Solving for stationary points for questions of the following type

How do you solve questions like $f(x,y) = x^2y + y^3x -xy$ for stationary points? A link to an educational resource that goes over this would be very helpful as well, as I don't even know what ...
1
vote
0answers
18 views

Calculating Hydrodynamic Interaction Tensor

I'm a bit of a newbie when it comes to Tensor calculus. Please excuse me as I learn... Given the Oseen tensor, $\mathbf{T}(\mathbf{R}) = (8\pi \eta R)^{-1} \left[ \mathbf{I} + ...
1
vote
1answer
32 views

Geometric interpretation of ${\partial f\over \partial x}= {\partial f \over \partial y}$

I know that $${\partial f\over \partial x}= {\partial f \over \partial y}$$ iff there exists a differentiable function $g$ (of one variable) such that $g(x+y)=f(x,y)$ (where $f : D\subseteq \mathbb ...
1
vote
0answers
30 views

Finding maximum value of a 3-variable function using inequality.

Let $a, b, c$ be positive real numbers satisfying $a^2 +b^2+c^2=14$. Find the maximum value of $f(a,b,c)=\frac{4(a+c)}{a^2+3c^2+28}+\frac{4a}{a^2+bc+7}+\frac{5}{(a+b)^2}-\frac{3}{a(b+c)}$
0
votes
2answers
37 views

Finding image and inverse function in $\mathbb R^2$

Let $V \subseteq \mathbb R^2$ be open subset which is surrounded by lines: $y=x;~ y=2x;~ x+y=1;~ x+y=3$ and $m:= \frac{y}{x},~ s:=x+y$ for $(x,y) \in V$, so it follows that $y= \frac{ms}{m+1}, ~ x= ...
0
votes
2answers
26 views

find the derivative of a function with more than one variable

I have a function $g(a)=f_i(x+a(y-x))$ where a$\in$$\Re$ and x,y$\in \Re^d$. How can I find the first and the second derivative of this function? The second part of the exercise is asking me to use ...
2
votes
1answer
61 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 ...
0
votes
0answers
18 views

How to derive the required Maxwell relation given that the constant term is not the same?

Question: What is the correct (and mathematically rigorous way) to derive the required Maxwell relation to complete the derivation of "Equation (1.1) becomes" in the text below? (Source: ...
0
votes
0answers
17 views

For which values of lambda is the set of line integrals bounded above?

Let P = {(x,y,z) $\in$ $R^3$ | 0$\le$ z$\le$1, 1$\le$$x^{2}$+$y^2$$\le$4}. For $\lambda$$\in$R, consider the vector field $$F_\lambda(x,y,z) = (2x+ \lambda y,-\lambda x+2y,2z) $$ in P. For which ...
0
votes
1answer
12 views

Finding linear equation for the plane of equidistant point

I'm trying to do some practice questions in my book and I encountered this question: Find a linear equation for the plane consisting of all point $(x,y,z)$ in space that are equidistant from the ...
1
vote
2answers
23 views

Evaluate the integral along the stated curve

$\int_{C}{(3x+2y) \, dx + (2x-y) \, dy}$ along the curve y = sin($\pi*x\over2$) from (0,0) to (1,1). (Given that the curve is smooth). Approach: I attempted this problem by parametrizing x = ...
2
votes
0answers
29 views

The set composed of domain and codomain of integrable function measure zero

There is this problem which I have constructed a plan to prove, and I am stuck. If anyone could see my plan and tell what is wrong about it I would be very thankful. Let $f: Q \to [0,1]$ be ...
2
votes
3answers
29 views

Find $b$ so that $f(x,y) = y^3+3x^2y-15y-12bx$ has some critical point

I am trying to solve this excersice but I can't seem to get to anything but dead ends. Let $b\gt 0$ and $f(x,y) = y^3+3x^2y-15y-12bx$, find all possible values of $b$ so that $f$ has at least one ...
0
votes
1answer
33 views

Verify Green's Theorem for region bounded by the lines $x=2$, $y=0$, $y=2x$

Verify Green's Theorem for the region D bounded by the lines $x=2$, $y=0$, $y=2x$ and the functions $f(x,y)=(2x^2)y$, $g(x,y)=2x^3$. I have been trying this question for far too long and I can't ...
1
vote
1answer
22 views

Simplifying Double Integrals to Single-Variable Integrals

Let D be a subset of $\mathbb{R}^2$ defined by $ |x| + |y| \leq 1$, and let $f$ be a continuous single-variable function on the interval $[-1,1]$. Show that $$ \iint\limits_D \,f(x+y) \, \mathrm{d}x ...
1
vote
1answer
47 views

An example of a continuous function on $\mathbb R^2$ with two critical points, both of them minima

Knowing you can not use the minimum bound, there exists a function $f ( x , y )$ continuous in $\mathbb R ^ 2$ that has exactly two critical points which are (both) the minimum? Can you give me an ...
1
vote
2answers
24 views

Finding $ \dfrac{\partial z}{\partial x} \text{and}\dfrac{\partial z}{\partial y} $ if $ F(cx - az, cy-bz) = 0 $

If it is given that $ F(cx - az, cy-bz) = 0 $, then find $ \dfrac{\partial z}{\partial x} \text{and}\dfrac{\partial z}{\partial y} $ How do I go about doing this? I don't really understand which ...
0
votes
3answers
86 views

Can a non-zero vector field have zero divergence and zero curl?

I don't see how. Curl and divergence are essentially "opposites" - essentially two "orthogonal" concepts. The entire field should be able to be broken into a curl component and a divergence component ...
1
vote
0answers
20 views

Integral on sphere and ellipsoid

Let $a,b,c \in \mathbb{R},$ $\mathbf{A}=\left[\begin{array}{*{20}{c}} \mathbf{a}&{0}&{0}\\ {0}&\mathbf{b}&{0}\\ {0}&{0}&\mathbf{c} \end{array}\right]$ , det A $>1$ Let D = ...
0
votes
1answer
26 views

How to take the triple integral of $ \iiint_G xy\sin (yz)dV$

Hi I'm trying to evaluate $$\iiint_G xy\sin(yz) \ dV$$ where $G$ is the rectangular box defined by the inequalities $0 ≤ x ≤ \pi, 0 ≤ y ≤ 1, 0 ≤ z ≤ \pi/6$. I wasn't sure where to go after the first ...
1
vote
1answer
32 views

Intuition behind Laurent's theorem?

Taylor series has a pretty nice intuitive explanation. If you know the position, velocity, acceleration and so on of a particle you can predict it's location at any time. Does a similar intuitive ...
2
votes
1answer
73 views

$f: \mathbb{R}^2\to \mathbb{R}^2$ is differentiable, and satisfies an inequality that involves its partials - show that f is a bijection.

Suppose that $f: \mathbb{R}^2\to \mathbb{R}^2$ is differentiable, and the partial derivatives of the components $f_1$, $f_2$ satisfy $$max(|\frac{\ df_1}{dx} -1|, |\frac{df_1}{d_y}|, ...
0
votes
1answer
17 views

Prove that both iterated integrals exists but $f$ is not integrable

I need to prove that the function $f$, given by: $$f(x,y)= \begin{cases} 1 \iff (x,y) =(\frac p {2^n},\frac q {2^n}): (p,q,n) \in \Bbb N^3, 0<p,q<2^n \\0 \iff (x,y) \neq (\frac p {2^n},\frac q ...
2
votes
3answers
54 views

Strange double integral

What is wrong with this computation of $\int_0^1\int_{-y}^y \sqrt[3]{x} \, dx \, dy$? I'm considering real functions only. Since $x^{4/3}$ is an antiderivative of the integrand, we will get ...
3
votes
0answers
49 views
+50

Geometric Interpretation of Antiderivative?

Could someone please give me a geometric interpretation of the above theorem?
5
votes
1answer
85 views
+50

Geometric interpretation of analyticity?

Suppose the real valued functions $u(x,y)$ and $v(x,y)$ are continuous and have continuous first order partial derivatives in a domain $D$. If $u$ and $v$ satisfy the Cauchy Riemann equations at ...