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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2answers
30 views

Easy question regarding multivariable calculus

We know $f: \mathbb{R}^n \to \mathbb{R}^m$ is differentiable at $a$ if there exists linear $T$ such that $$ \lim_{h \to 0} \frac{f(x_0 + h) - f(x_0) - T(h)}{||h||} = 0$$ Why does this imply that ...
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1answer
44 views

Existence of a limit does not imply differentiability

I want to show that if the limit(in the definition of the derivative for a function $f: \mathbb{R}^d \to \mathbb{R}^m$) exists, then we do not necessarily have differentiability. My try: Consider ...
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1answer
46 views

if a function is linear, then it is its own derivative

LEt $f: U \subseteq \mathbb{R}^d \to \mathbb{R}^m$, $U$ open. Suppose $f$ is a linear transformation, and differentiable at $x_0 $. Then, my question is, do we have have that $Df(x_0) = f $ ?
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2answers
70 views

Definition of differentiability for multivariable calculus

In multivariable calculus, we say $f: \mathbb{R}^d \to \mathbb{R}^m$ is differentiable at $x_0 \in U \subseteq \mathbb{R}^d $, $U$ open if there exists a linear map $T: \mathbb{R}^d \to \mathbb{R}^m$ ...
5
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1answer
95 views

A line integral

If $\mathbf{B}(\mathbf{x})=\rho^{-1}\mathbf{e}_{\phi}$ in cylindrical polars, find: $$\int_{C}\mathbf{B}\cdot\mathrm{d}\mathbf{x}$$ where $C$ is the circle $z=0,\rho=1,\;0\leq \phi\leq ...
2
votes
1answer
75 views

Calculate the integral of a 2 form

I am trying to compute the integral $$ \int\int_{S}\frac{1}{x}dy\wedge dz+\frac{1}{y}dz\wedge dx+\frac{1}{z}dx\wedge dy $$ over an ellipsoid given by $$ ...
2
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1answer
130 views

Applying constrained maxima/minima and Lagrange Multipliers with eigenvectors and eigenvalues?

Let $A$ be a nonzero symmetric $3\times3$ matrix. Consider the function $f(\textbf x)=\frac{1}{2}(A \textbf x)\cdot \textbf x$. (a) What is $\nabla f$? Here's what I did: Let $A= \left[ ...
2
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2answers
365 views

Demonstrate that the limit of a function of two variables does not exist

From my multivariable textbook: $$\lim_{|x,y|\to|0,0|}\frac{y^2\sin^2 x}{x^4+y^4}$$ (original screenshot) Wolfram indicates that the limit DNE, but does not list the steps used to solve. Is there ...
2
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1answer
89 views

Derivative of a Vector with respect to its norm (special relativity)

I came across an equation (related to special relativity) that requires me to to take a derivative of a vector with respect to to it's own norm. In a bit more detail, what I mean is, let: $$\vec ...
0
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2answers
99 views

Supremum Definition in context of a Rectifiable/Differentiable Curve

I am trying to prove that if I have 2 paramterizations of the same curve $\gamma$ and $\sigma$ (i.e. there is continuous bijective map $\phi$ such that $\sigma = \gamma \circ \phi$) then if the curve ...
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1answer
37 views

Finding the curl - what am I doing wrong?

I must be making a really daft mistake. If $\mathbf{F}=-y\mathbf{e}_x+x\mathbf{e}_y=r\mathbf{e}_{\phi}$ (in Cartesian and Cylindrical polars respectively): ...
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2answers
73 views

Differentiability of a multivariable function

I want to study differentiability of $f$ at the origin: $$ f(x,y) = ( x^3 + y^3)^{1/3} $$ MY attempt: I claim $f$ is not differentiable at the origin because its partial derivatives are not defined ...
3
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0answers
134 views

Understanding the setup for the probability that $Ax^2+Bx+C$ has real roots if A, B, and C are random variables uniformly distribted over (0,1).

Suppose that $A, B,$ and $C$ are independent random variables, each being uniformly distributed over $(0,1)$. What is the probability that $Ax^2 + Bx + C$ has real roots? First, I set $P(B^2 - 4AC ...
1
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1answer
110 views

Partial derivative on convex set

If we have a function $f:U \rightarrow R$ ($U \subset R^n$) which is partially differentiable on a convex set U with $\frac{\delta f}{\delta x_1} = 0$ for all $x \in U$. How can we prove that $f$ ...
0
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2answers
35 views

multivariable calculus - find a function such that $\lim_{t \to 0} f(tx,ty)=0$

I was asked the following question: Find a function $f:\mathbb R^2 \to \mathbb R$ such that $f(x,y)$ has no limit as $x$ and $y$ approach zero, but $\lim_{t\to 0} f(tv)=0$ for all $v \in \mathbb R^2$ ...
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1answer
95 views

How to obtain a diffeomorphism between $\mathrm{SL}(2,\mathbb{R})$ and $ (\mathbb{C}\!\smallsetminus\!\{0\})\times\mathbb R$?

Could someone please give me a tip on how to show that the map $\mathrm{SL}(2,\mathbb{R}) \to (\mathbb{C}\setminus\{0\})\times\mathbb{R}$ \begin{equation}\begin{pmatrix} a & b\\ c & d ...
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0answers
52 views

Can the Euler-Lagrange equations be derived from a variation over a time of order $dt$ rather than $t$?

In the calculus of variations, the solution of the Euler-Lagrange equations gives those functions for which a given functional is stationary. Now all derivations I've come across up to now, carry out ...
2
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1answer
80 views

Differentiability in $R^n$

I have the definition of the derivative for $f:\mathbb R^n \rightarrow\mathbb R^m$ at a point $a$ as: $f$ is differentiable at a then there exists a linear map $L:\mathbb R^n \rightarrow\mathbb ...
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1answer
48 views

A characterization of convexity for functions with vectors as domain.

Let $f:\mathbb R^n \rightarrow \mathbb R$ be a continuously differentiable function. By $df(w)$ I denote the Frechet derivative of $f$ at $w$ Prove that $$f \:\text{is convex} \Leftrightarrow ...
2
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1answer
51 views

Volume with triple integrals

Calculate integral $$\iiint_V \frac{e^{-x^2-y^2-z^2}}{\sqrt{x^2+y^2+z^2}} dV$$ Where $V\subset\mathbb{R}^3$ is the exterior of a origocentered sphere with radius of 2 \begin{align*} V=&\iiint_V ...
0
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1answer
143 views

Derivative of a function along a path

I'm having problems visualizing and understanding the chain rule for the derivative of a function along a path. i.e $$\frac{df(r(t))}{dt}=\nabla f(r(t))\cdot r'(t)$$ My thinking is that this result ...
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1answer
43 views

Centre of mass question

Find the centre of mass $\overline{P}=(\overline{x},\overline{y},\overline{z}) $ of a unconstrained body $0\le z \le e^{-(x^2+y^2)}$. The density $\delta(x,y,z)$ of the body is constant. I think we ...
2
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3answers
71 views

Vector valued function in $\mathbb{R}^2$ with non-finite arc length

Find a curve $\mathcal{C}$ : x = g(t) , $a \leq t \leq b$ in $\mathbb{R}^2$, where g $\in \mathcal{C}[a,b]$ such that $\mathcal{C}$ does NOT have finite arc length
3
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1answer
83 views

Prove $\nabla f$ is orthogonal to the surface $f$

I'm trying to prove that $\nabla f$ is orthogonal to the surface $f$. I think I have a valid proof but I'm not sure that it is rigorous. To prove $\nabla f$ is normal I am proving that $\nabla ...
2
votes
2answers
119 views

Solving a curve integral around part of an elipse

I'm having trouble calculating a curve integral in a vector field: $\int_C y (18x + 1)\ dx + 2y^2\ dy$ where $C$ is the curve along the ellipse $9x^2 + y^2 = 64$ going counterclockwise from the ...
0
votes
2answers
61 views

By Lagrange Multipliers, the function $f$ has no minima or maxima under constraint $g$?

Find the extrema of $f$ subject to the stated condition. $f(x,y)=x-y$ subject to $g(x,y)=x^2-y^2=2$. Ok, by Lagrange Multiplier method, we find the points that satisfy $\nabla f(x,y) = \lambda ...
0
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3answers
88 views

Calculating a triple integral

I'm given with the region $V$ , determined by the following surfaces: $x^2 =y , y^2 =x ,z=0 , z=1$ and need to calculate: $\iiint _V \frac{\sin x-\sin y}{xy+1} dx\,dy\,dz $ and to use symmetry. I ...
2
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1answer
55 views

Harmonic map into sphere

Let $B$ be the unit ball and $S$ the unit sphere in $\mathbb{R}^3$. Consider the map $u: B\rightarrow S$ defined as: \begin{equation} u^j(x)=\frac{x_j}{|x|}\quad\forall \ j =1, 2, 3. \end{equation}I ...
0
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1answer
58 views

how to solve this homogeneous differention equation ?

$dy/dx= (2x+3y+4)/(4x+6y+5)$. I am trying to solve this homogeneous Ds, but don't understand how to solve it. I believe the first step is to solve this: 1) $y=u x$ 2) $dy/dx = u+x \cdot du/dx$ then ...
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3answers
109 views

Surface integral

Problem statement $$ \mbox{Calculate the surface integral}\quad \int_{Y}\ y\,\sqrt{z\,}\,\sqrt{4x^{2} + 4y^{2} + 1\,}\,\,{\rm d}S $$ where $Y$ is the surface $\left\{\left(x,y,z\right)\ \ni\ ...
0
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1answer
57 views

If a function is complex differentiable, how do we know its real and imaginary parts are infinitely differentiable?

Sorry I'm really rusty on multivariable calc. Suppose $f: \mathbb{C} \to \mathbb{C}$ is holomorphic and $f(x,y) = u(x,y) + v(x,y)i$, then we know that the partials $u_x, u_y, v_x, v_y$ exist and are ...
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2answers
28 views

Variance of three possible outcomes

I am new to this kind of things so maybe you could help me get the reasoning. I have a continuum of outcomes on the interval $[0,1]$. Now, let us cut the interval into two pieces so that there are two ...
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1answer
69 views

How to show $\int_Q \varphi^2 (\Delta v)v_t = \int_Q |\nabla v|^2 \varphi \varphi_t - 2\int_Q (\nabla v \cdot \nabla \varphi) (\varphi v_t)$

Let $Q=\bigcup_{t \in (0,T)}\Omega \times \{t\}$. I have seen this identity for all $\varphi \in C_c^\infty(Q)$ such that $0 \leq \varphi \leq 1$: $$\int_Q \varphi^2 (\Delta v)v_t = \int_Q |\nabla ...
0
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1answer
14 views

If $f $ is $C ^1 $ does it follow that $H(\textbf {x})= f(\textbf {x}) - E \cdot \textbf {x}$ is $C^1 $?

Let let $f $ be of class $C ^1 $ and define $E = D f(\textbf {a}) $. Now define $$ H(\textbf {x})= f(\textbf {x}) - E \cdot \textbf {x} $$ Does it follow that $H $ is of class $C ^1 $?
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1answer
27 views

Multivariable Integral-Polar

Find the volume of the spherical wedge above $ z = r \cot \phi_o$ and below $z^2 + r^2 = \rho_0^2$, and on the sides by $\theta_1$ and $\theta_2$. I tried to set an iterated integral ...
0
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1answer
143 views

If the partial derivatives are continuous then the tangent plane exists.

Background: I am studying Calculus from Stewart which is not rigorous. So, Stewart does not talk about this problem. Also, multivariable calculus is dealt mostly in context of three dimensions ...
0
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1answer
186 views

How to find the direction in which a function does not change at a point?

I have a function $f(x, y)$ and a point $P = (x_{0}, y_{0})$. I need to find the direction $\vec{u} = (a, b)$, among four options, in which the functon $f$ does not change. How can I do that? UPDATE: ...
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1answer
52 views

Partial differentiation of a composite function

This should be straightforward, but don't seem to be able to crack it. Take a function $f(x_1, x_2, x_3)$ and a function $g(x4, x5, x6)$. These two functions mapp from $R^3 \rightarrow R^1$. I am ...
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1answer
34 views

Line Integral using Green's theorem

Problem statement: Calculate $\int_{\gamma }(3e^{(y-3x)^{2}}-y)\mathrm{dx}+(-e^{(y-3x)^{2}}+2x)\mathrm{dy}$ where $\gamma$ is the curve $y=x^2$ from $(0,0)$ to $(3,9)$. Progress First idea was to ...
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2answers
178 views

An implicit function theorem question

In my course on multivariate calculus we treat the implicit function theorem and I am stuck on the following question: Find the values of $a$ and $b$ such that, in a neighbourhood of $(x,y,u,v) = ...
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0answers
88 views

What is difference between directional derivative and gradient?

What is difference between the directional derivative $\frac{df(r)}{dr}$ and the gradient $\nabla f(r)$ of a function $f$?
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1answer
50 views

is this intengral bounded?

Im just stuck beacause I dont know if this integral in bounded, I was trying to make a change of variable but I cant get to anything: (edited what need is that f is bounded for a fixed x) ...
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4answers
1k views

Calculator similar to Desmos but for 3D

Is there a calculator with functionality similar to Desmos but in 3 dimensions? I am looking to learn about families of quadric surfaces so I am looking for a 3D calculator with sliders.
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3answers
229 views

Using Lagrange multipliers to solve for minimum

I am having troubles with one part of this homework problem. Hopefully somebody can help me out: Find the minimum and maximum values of the function $f(x,y)=x^2+y^2$ subject to the given constraint ...
0
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1answer
92 views

Sign of Laplacian at critical points of $\mathbb R^n$

Suppose we are in $\mathbb R^n$. What can we say about the sign of $\Delta u(\vec x)$ if u($\vec x$) has a local max/min at $\vec x$? I've tried looking at the reverse of the second partial derivative ...
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2answers
484 views

Find partial derivatives of function f(x,y) where f(x,y) is defined as an integral

<. Find the partial derivatives <(x,y)< and <(x,y)<, in terms of the function <, where the function < is defined by the following integral. <$$ f(x,y)=\int_a^bh(t) dt ...
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1answer
38 views

Transposition $s=4t+\ln(1−t)$ [closed]

At any time t secs the distance s metres of a particle moving in a straight line from a fixed point is given by: $$s=4t+\ln(1−t).$$ Determine: a) The Initial Velocity and Acceleration b) The ...
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0answers
49 views

Getting “semi” orthogonal basis from a linear independent set

Let $K_i: \mathbb{R}\mapsto \mathbb{R}^k$ are continuous functions for all $i=1,\dots,k-d$ such that for every fixed $t\in\mathbb{R}$ we have ${\cal K}_t=\{K_1(t),\dots,K_{k-d}(t)\}$ be a linear ...
0
votes
3answers
190 views

A doubt on a multivariable calculus result

Let $f: U \subseteq \mathbb{R}^d \to \mathbb{R}$ be differentiable. $U$ open. Is it true that for any $v = ( \alpha_1,\dots,\alpha_d ) \in \mathbb{R}^d $, we have $$ D_v f(x_0) = \sum_{j=1}^d ...
0
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2answers
155 views

Question about the differential

Today at class, my teacher stated the following proposition saying it is obvious: Let $x_0 \in U \subset \mathbb{R}^d$, $U$ open, and $f: U \to \mathbb{R}^m$ differentiable at $x_0$, then for any $v ...