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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-1
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0answers
18 views

how to check partials derivatives are continuity? [on hold]

If a function has partials how to check that the partials are continuous ? And how on a piecewise function?
0
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0answers
13 views

how to prove a multivariable function is differential? is another procedure

How to prove a multivariable function is derivable? Is to see if a function has some points of discontinuity by inspection and check on every point of possible discontinuity using the definition of ...
0
votes
1answer
18 views

Vector calculus identities

Let $f$ be scalar potential for the vector field $\underline u $ (i.e $\underline u = -\underline \nabla f$). Prove that the vector field $$ \underline r \wedge \underline u $$ has magnetic ...
0
votes
1answer
32 views

Gradient of real part = real part of gradient?

Suppose f(x,y,z) maps $\mathbb{R}^3\rightarrow\mathbb{C}^1$. That is, it takes in three real numbers and spits out a complex number. Does the following always hold: $$\vec\nabla ...
0
votes
1answer
14 views

why $f|_{U}$ is immersion for some open set containing $a$

Its written that if $f$ is immersion at $a$ then $f|_{U}$ is immersion for some open set containing $a$. I don't understand why its happening.. can one please explain ?
1
vote
2answers
37 views

Find the volume below $\sqrt{x}+\sqrt{y}+\sqrt{z}=1$ in the first quadrant

I understand that we have to use transformation $$x = u^2, y = v^2, z = w^2$$ but I cannot figure out the limits. I just need a rough sketch of how to approach this. Could anyone give me some ideas?
0
votes
2answers
53 views

Find the smallest value of $f(x, y, z)$

Find the smallest value of $f(x, y, z) = \sqrt{x^2 + 1} + \sqrt{(y - x)^2 + 4} +\sqrt{(z - y)^2 + 1} + \sqrt{(10 - z)^2 + 9}$ I found this question while looking from some exam papers and have no ...
2
votes
2answers
47 views

What is the order of the PDE $\newcommand\pp\partial\frac{\pp^2u}{\pp x^2}+\frac{\pp^3u}{\pp x^2 \pp y}+\frac{\pp^2u}{\pp^2y}=xy\frac{\pp u}{\pp x}$? [on hold]

The order of the differential equation $$\frac{\partial^2 u}{\partial x^2}+\frac{\partial^3 u}{\partial x^2 \partial y}+\frac{\partial^2 u}{\partial^2 y}=xy\frac{\partial u}{\partial x}$$ is ...
0
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0answers
61 views

How to prove partials derivatives are continuous?

To prove that a functions has partial derivatives every partial has to exist, and every partial exist only if the limit of definition of partial exist. Is this right? Then if you have partials and ...
0
votes
0answers
13 views

Deduction of the equation of continuity in one dimension

I need the deduction of the continuity equation in one dimension usig the result: $$\frac{\partial}{\partial t}\int^{b(t)}_{a(t)} \rho(x,t)dx=\rho(x,t)b'(t)-\rho(x,t)a'(t)+\int^{b(t)}_{a(t)} ...
0
votes
0answers
11 views

parametrization of intersecting level curves in neighborhood of given point

Let $f(x,y,z) = yarctan(x) +z^2,g(x,y,z) = xy^2 + xyz + z $ and let $\gamma$ be the intersection curve between the surfaces $f(x,y,z) = 1$ and $g(x,y,z) = 1$ Show that $\gamma$ can be ...
0
votes
0answers
17 views

Fréchet normal cone

Given $x\in \Omega(\subset X)$ (X: Banach space) and $\varepsilon\geq 0$, the set of $\varepsilon-$normals to $\Omega$ at $x$ by \begin{align} \widehat N_\varepsilon(x;\Omega):=\left\{x^*\in X^*\mid ...
1
vote
3answers
22 views

Check if two vector equations of parametric surfaces are equivalent

Give the vector equation of the plane through these lines: $\begin{pmatrix}x\\y\\z\end{pmatrix}=\begin{pmatrix}4\\1\\1\end{pmatrix}+\lambda\cdot\begin{pmatrix}0\\2\\1\end{pmatrix}\,\,\,$ and ...
0
votes
1answer
25 views

Continuity of the maximum of a function in two variables

The function $f( x, y)$ is continuous on $x\in [a,b]$, $y\in [a,b]$. Is the function $g(x) = \max_{y} f( x, y)$ continuous on $x\in [a,b]$?
0
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0answers
27 views

Solving the Telegraph Equation using Partial Differential Equations and Sturm-Liouville theory

I've been asked to do the following question, and I've got through the brunt of it (so this is going to be a rather long question...), but I'm just having a bit of trouble applying Sturm-Liouville ...
0
votes
1answer
19 views

Volume of a solid in spherical coordinates

How might we find the volume of the solid whose surface is $\rho = \sin{\phi}^{1/3}$? Of course, the obvious way to proceed is to write the triple integral $$\int_V dV$$ taking of course $dV = ...
-1
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0answers
34 views

Partial derivatives of a piecewise defined function

If a function f as $$f(x,y) =\begin{cases} x^2+2x+5y+10 & \text{ for } (x,y)\neq (0,0) \\ y^2+2y+x+10 & \text{ for } (x,y)=(0,0) \end{cases}$$ Is it true that $$f^\prime_x(x,y) ...
1
vote
1answer
23 views

How can I numerically evaluate the total derivative of a multivariate function?

I think I understand now the intuitive reasoning behind the total derivative of a multivariate function $z = z(x, y)$, which is $$ dz = \frac{\partial{z}}{\partial{x}}dx + ...
1
vote
1answer
26 views

Confusion about the Total Derivative

I just started multivariable calculus a little while ago and I'm confused about the concept of a total derivative of some function $z = z(x, y)$. I was taught that $dz = \frac{\partial z}{\partial ...
1
vote
2answers
29 views

no diffeomorphism from $\mathbb{R}^2 \to \mathbb{R}^3$

Show there is exist no diffeomorphism from $\mathbb{R}^2 \to \mathbb{R}^3$ PS: Don't say $\mathbb{R}^2,\mathbb{R}^3$ aren't homeomorphic, I need explanation without using topology
2
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0answers
22 views

Normal Vector Affecting The Divergence Theorem

$\newcommand{\Div}{\operatorname{Div}}$I'm going to use an example to explain what I'm trying to ask. Let $T =\{(x,y,z): x^2+y^2=z^2, 0\leq z\leq3\}$, I'm asked to calculate $\iint_T ...
0
votes
0answers
15 views

Firm non-expansiveness in the context of proximal operators

$\newcommand{\prox}{\operatorname{prox}}$ Probably the most remarkable property of the proximal operator is the fixed point property: The point $x^*$ minimizes $f$ if and only if $x^* = \prox_f(x^*) ...
1
vote
1answer
13 views

line integrals and partial derivatives statement (Green's theorem application)

Let $P(x,y),Q(x,y)$ be $C^1$ functions of $\mathbb R^2$, prove that the following statements are equivalent: (1) $P_x-Q_y=0$ and $P_y+Q_x=0$ (2) For every simple closed curve $C$, it is satisfied ...
1
vote
1answer
22 views

geometric interpretation of the norm: $\|\vec x\|={(|x_1|+|x_2|)\over 3}+{2\max(|x_1|,|x_2|)\over 3}$

Let $p:\mathbb R^2 \to \mathbb R$ be a norm so that $$\|\vec x\|={(|x_1|+|x_2|)\over 3}+{2\max(|x_1|,|x_2|)\over 3}$$= $${{\|\vec x\|_1\over 3}}+{2\|\vec x\|_\infty\over 3}$$ The thing is that I need ...
4
votes
1answer
37 views

Mixed partial derivatives are different

Let $f: \Bbb R^2 \to \Bbb R$ be defined as $$f(x) = \left\{ \begin{matrix} x_1^2 \operatorname{arctan} \left( \frac{x_2}{x_1} \right) - x_2^2 \operatorname{arctan} \left( \frac{x_1}{x_2} \right), ...
1
vote
1answer
40 views

Calculating multi-variable limit.

I am struggling to find a way to approach this limit $$\lim_{(x,y)\to(0,0)}\frac{\sin(x^2y+x^2y^3)}{x^2+y^2}$$ I would greatly appriciate if You could explain to me how to solve it or at least show ...
2
votes
1answer
20 views

continuity single and multivariable function simple question

Why $$f(x,y) =\begin{cases} \frac{xy^2}{x^2 +y^2} \mbox{ for } (x,y)\neq (0,0) \\ 0 \mbox{ for } (x,y)= (0,0)\end{cases}$$ is continuous and $$f(x) =\begin{cases} 2 \mbox{ for } 0>=x>10 \\ 5 ...
0
votes
1answer
20 views

continuity single variable function and multivariable funtion and its parcial derivatives

Is f(x)=1/x discontinuous at point x=0 or not since its domain is x>0 and x<0? And what about f(x,y)=$\frac{xy^2}{x^2+y^2}$ continuity? And Df(x,y) exist or parcial derivatives are ...
2
votes
1answer
21 views

Surface Integral over a sphere

Suppose $f(x,y,z)=g\left(\sqrt{x^2+y^2+z^2}\right)$, where $g$ is a function of one variable such that $g(2)=-5$. Evaluate $$\iint_S f ~dS,$$where $S$ is the sphere $x^2+y^2+z^2=4$. Now, I ...
0
votes
1answer
28 views

Optimization with a constraint given by a differential equation

I have the following differential equation $$\ddot\theta(t) = -k\sin{\omega t}\sin{\theta(t)} \quad \text{where} \quad \theta(0)=\theta_0, \dot\theta(0)=v_0$$ where $\omega$ is a known constant and ...
-7
votes
0answers
60 views

Can a set in $\mathbb{R}^2$ be closed but unbounded?

Today I read "on a closed, bounded set $D$". How can a set be closed but not bounded?
0
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0answers
26 views

Sufficient Conditions for Multivariate Decreasing Function

I found the following helpful theorem concerning decreasing functions but it's only valid for $\varphi:\mathbb{R}\rightarrow \mathbb{R}$, I'd like to know if it can be extended to the ...
-2
votes
1answer
34 views
0
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0answers
35 views

Book on calculus of several variables.

I'm an undergraduate student in mathematics and want to study Calculus of several variables currently this semester which involves the use of analysis,vector spaces and linear transformations. Can ...
2
votes
2answers
67 views

Which book is appropriate for a Chemistry student that needs to learn basics about integrals?

A friend of me who is not studying mathematics now needs to deal with integrals, double integrals and triple integrals within his study of chemistry. He asked me to give him a suggestion for a basic ...
0
votes
1answer
25 views

The meaning of product of functions in multivariable calculus

If $f$ and $g$ are $2$ functions $\mathbb{R}^n\rightarrow\mathbb{R}^m$ For $m=1$ and $n>1$ is $f\cdot g$ or $(f\cdot g)(x)$ defined for? Would that be a real number? And for $n=1$ and $m>1$, is ...
0
votes
0answers
42 views

Arc length for a function $f:\mathbb{R}^2 \to \mathbb{R}^2.$

Assume $f:\mathbb{R}^2 \to \mathbb{R}^2$ is $C^1.$ Is there a formula for the length of the subset of $\mathbb{R}^2$ given by $$ \{f(x,y) \in \mathbb{R}^2:a_1\leq x\leq b_1, a_2\leq y \leq b_2\} ? $$ ...
0
votes
1answer
43 views

Integral of a bivariate normal cdf

Let $$ \Phi_2(x,y;\rho):=\int_{-\infty}^y\int_{-\infty}^x \frac{1}{2\pi\sqrt{1-\rho^2}}e^{-\frac{1}{2(1-\rho^2)}(s^2+t^2-2st\rho)} \, ds \, dt $$ be the joint cdf of bi-variate normal random ...
2
votes
2answers
43 views

Find the work done by the force field in moving the particle from one point to another

Find work done by the force field F in moving the particle from $(-1, 1)$ to $(3, 2)$ This sounds good till we are given that $\textbf{F} = \dfrac{2x}{y}\textbf{ i }- \dfrac{x^2}{y^2}\textbf{ j }$ ...
5
votes
2answers
67 views

When is $\min_{x\in X,y\in Y}f(x,y)=\min_{x\in X}(\min_{y\in Y}f(x,y))$?

When is $$ \min_{x\in X,y\in Y}f(x,y)=\min_{x\in X}(\min_{y\in Y}f(x,y))? $$
6
votes
2answers
59 views

Continuity of $\frac{2xy}{x^2+y^2}$ at $(0,0)$

Given a Heaviside function $$f(x,y)=\begin{cases}\frac{2xy}{x^2+y^2}, &x^2+y^2 \neq 0\\0 ,&x^2+y^2=0 \end{cases}$$ Letting $a$ and $b$ be fixed constants, show that for all values of ...
0
votes
1answer
25 views

Let $V$ be the space of real sequences {${x_{1},x_{2},…}$} so that $\sum_{k=1}^\infty {x_{k}}^{2}$ converges. Prove that this space is not numerable

Let $V$ be the space of real sequences {${x_{1},x_{2},...}$} so that $\sum_{k=1}^\infty {x_{k}}^{2}$ converges. Prove that this space is not numerable: My attempt: I have already proved that this is ...
10
votes
1answer
239 views

Intuition behind curl identity

Is there any clear intuition behind the identity $$ \nabla\times (\nabla\times A)=\nabla (\nabla \cdot A)-\nabla ^2A $$ Though the result is useful and not difficult to derive, it doesn't quite ...
3
votes
1answer
31 views
+50

Rewriting a continuously differentiable function

I have the following $i$-th regressor function: $\phi_i(x)$ in which $x$ is a vector with elements $x_1, \ldots, x_n$. I cite from an article: Let $e_i = \hat{x}_i - x_i$ and note that, since ...
-1
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0answers
35 views

For a map $f :\mathbb R^m\to\mathbb R^n$, prove that $\lim_{x\to a} f(x)=L$ if and only if $\lim_{x\to a}\|f(x) − L\|=0$

Could anyone help me with this proof? Given a map $f : \mathbb R^m \to\mathbb R^n,$ prove that $\lim_{x \rightarrow a} f(x) = L$ holds if and only if $$\lim_{x ...
1
vote
1answer
24 views

Calculating moment of inertia about the $z$-axis of solid with constant density

I have the following math problem: Find the moment of inertia about the $z$-axis of the solid in the first octant that is bounded by the coordinate planes and the graph of $x+y+z=1$ if the density ...
0
votes
3answers
23 views

Finding a vector normal to the plane with position point and parallel to two vectors

A cross product of the two parallel vectors will get the vector normal to the plane. But I'm looking for a specific normal vector. The plane with point $A(3,2,1)$ and parallel to $u = 5i+2k$ and $v= ...
0
votes
1answer
16 views

Does the map $F(x, y)=(f(x, y), y)$ induce an isomorphism $dF_{(a, b)}$?

Suppose $f:\mathbb R^{n+m}\longrightarrow \mathbb R^n$ be a $C^p$ map such that $df_{(a, b)}:\mathbb R^{n+m}\longrightarrow \mathbb R^n$ is surjective for some $(a, b)\in\mathbb R^{n+m}$. Define ...
2
votes
3answers
26 views

Gradient of modulus of vector.

I came across this in my lecture notes: This is using index notation, non-bold r is the modulus of r, and the partials are with respect to the components of r. I understand most of the steps, but ...
0
votes
0answers
18 views

Multivariable chain rule and an integral function

Can the chain rule be used in the following scenario? Let $u(r,t) = \int g(r,t)~f_R(r)~dr$ where $f_R(r)$ is a well-defined probability distribution. Then we can write $u(r,t)=u(x,y)=\int x~y~dr$ ...