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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11 views

Existence of some functions

Let $\omega = \frac{(x+1)dy -y dx}{(x+1)^2+y^2}-\frac{(x-1)dy-y dx}{(x-1)^2+y^2}$. Show that there is a function $f: \mathbb{R}^2 -[-1,1]\times \{0\} \longrightarrow\mathbb{R} $ such that ...
2
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1answer
33 views

Showing that $f_{xy} = f_{yx}$ for the following function.

Show that for the function $$f(x,y) = 9x^3y+2y^3+10x^2y^2+9$$ satisfies the equality $$f_{yx} = f_{xy}$$ by computing the partial derivatives. I know that $f_y= 9x^3+6y^2$ because we exclude all ...
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0answers
27 views

Interpreting notation for a differential equation

I'm looking at the following differential equation (see Appendix B of the following paper): $$ \partial_t\mu[V_E]=-\mu[V_E] + C_{EE}[x\nu_E+(1-x)v_{ext}]-C_{EI}J_{EI}\nu_I, $$ where $\mu[V_E]$ is ...
1
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1answer
15 views

Projection from triangle to spherical triangle

Consider a triangle, $T$, in $\mathbb{R}^3$ with vertices $(0,0,1), (0,1,0)$, and $(1,0,0)$. Let $S$ denote the sphere centered at the origin with radius 1 and let $S_1$ denote the portion of the ...
0
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0answers
11 views

Problem with composite functions

Let's say I know two bivariate functions: $f_1(x_1,x_2)$ and $f_2(x_1,x_2)$. Is there a standard way to find functions $g(x,y)$ and $h(x)$ such that: $(g\circ(f_1,f_2))(x_1,x_2) = (h\circ ...
0
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1answer
20 views

Existence of the partial derivatives ${\delta^{2}f}\over {\delta x \delta y}$ and ${\delta f}\over {\delta x}$

The question is can the partial double derivative ${\delta^{2}f}\over {\delta x \delta y}$ exist without the derivative ${\delta f}\over {\delta x}$ existing? I don't know , I am ...
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0answers
18 views

A form of chain rule to differentiate the flow of a vector field on a manifold

I am reading the proof for a theorem about connections on a manifold, but I'm not comfortable with the fancy language of vector bundles and flows of vector fields I think. I wonder if there's an easy ...
1
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1answer
27 views

Availability of derivative for multivariable functions: Are these conditions equivalent?

In Hubbard's multivariable calculus book a function $f:\mathbb R^m \mapsto \mathbb R^n$ has a derivative at a point $a \in \mathbb R^m$ if the following equation holds: $$\lim_{\vec h \to \vec ...
0
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1answer
10 views

Show that in Line Search Methods the “steepest descent direction” is the one along which the objective function decreases most rapidly

I want to verify the claim, that the steepest descent direction $-\nabla f(x^k)$ is the one along which $f\in C^2(\mathbb R^n)$ decreases most rapidly. Therefore, I considered the Taylor expansion ...
0
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1answer
37 views

Confusion about Partial Derivative for a Function of One Variable

This question actually came up as I was reading an example in my differential equations book (Boyce & Diprima): Solve: $2x+y^2+2xyy'=0$ Define $\psi(x,y)=x^2+xy^2$ Then ...
1
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1answer
25 views

Direction where the directional derivative is maximal

I am given the following function: $$ f(x,y)=\sqrt[3]{x^2 y } $$ at (0,0), and need to find the directions $\vec{v}$ for which the directional derivative $D_\vec{v} f (0,0)$ is maximal. I know the ...
0
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2answers
37 views

Question involving gradient of a function.

We are given any arbitrary ellipse with focii $F1$ and $F2$ , $T$ is the unit tangent to the ellipse through a point $P$. Let $f$ be the sum of the distances of the of $F1$ and $F2$ from $P$ , we ...
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1answer
18 views

Find local extrema of $x_1+5x_2$ when $2x_1+3x_2=1$ and $x_1-x_2+x_3=0$.

I'm trying to solve the following problem: Find all local extrema of $f:M\to\mathbb{R}$ where $$ M :=\left\{x\in\mathbb{R}^3: 2x_1+3x_2=1,\ x_1-x_2+x_3=0\right\} $$ and $$ f(x)=x_1+5x_2,\ ...
-2
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1answer
25 views

Calc 3 Problem about work [on hold]

Find the work done by the force → F = −2k to move an object from the point P(2, 1, 1) to the point Q(−1, −1, 1).
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0answers
7 views

Finding Conditional Distribution of Multivariate distribution

Q: Supposing y ~ N4 (µ,∑) and µ = (1 2 3 -2)' ∑ =\begin{pmatrix} 4 & 2 & -1 & 2\\ 2&6&3&-2\\ -1&3&5&-4 \\ 2 &-2 ...
1
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1answer
34 views

Find lower and upper bound of $f: \{x \in \mathbb{R}^3 : x_{1}^2 + 2x_{2}^2 + 3x_3^2 \le 6 \} \to \mathbb{R}$

$f$ is given by the formula : $f(x) = 2x_1 + 4x_2 - 6x_3$ Since the domain of $f$ is a bounded and closed set, $f(x)$ does have upper and lower bounds, either in the interior of its domain or on the ...
3
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1answer
33 views

Why does max. increase have to be along the x,y,z axis in gradient?

$$\nabla f = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}\right)$$ These components are the rate of increase along the $x$, $y$ and $z$ directions ...
2
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2answers
43 views

What is the matrix $\left[ DS(A) \right]$, which gives $\left[ DS(A) \right] H=AH+HA$?

In Hubbard's multivariable calculus book $DS(A):H \mapsto AH+HA$ is introduced as a linear transformation where $A$ is an $n \times n$ matrix, $S(A)=A^2$, and $D$ is the notation for derivative. It ...
0
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1answer
28 views

Can single variable function be represented by field? [on hold]

Is field concept in mathematics directly related to multi variable functions? Can single variable function be represented by field?
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1answer
30 views

Equality of mixed partial/total derivative

I have $F = F(x_1(t),x_2(t),\dotsc,t)$, where $x_1,x_2,...$ are (unknown) functions of $t$. Everything is continuous, differentiable, etc. Is it possibly, necessarily, or never true that ...
2
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3answers
110 views

$f:\mathbb R^{2} \rightarrow \mathbb R$ s.t ${f(x,y)}={{xy}\over {x^{2}+y}}$ is not continuous at the origin

$f:\mathbb R^{2} \rightarrow \mathbb R$ is defined as $${f(x,y)}={{xy}\over {x^{2}+y}}$$; when $x^{2}+y\neq 0$ and $$f(x,y)=0$$ otherwise. To show this is not continuous at the origin . ...
0
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1answer
19 views

If $A^k$ consistently approximates $\nabla^2f(x^k)$ with $x^k\to x^*$ and $\nabla^2f(x^*)$ regular, then the $A^k$ are regular, too

Let's call $\left\{A^k\right\}\subseteq\mathbb R^{n\times n}$ a consistent approximation of $\left\{B^k\right\}\subseteq\mathbb R^{n\times n}$ iff ...
0
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3answers
31 views

Trying to show $|\overrightarrow{a}\times\overrightarrow{b}|^2=|\overrightarrow{a}|^2|\overrightarrow{b}|^2-(\overrightarrow{a}⋅\overrightarrow{b})^2$

If $\overrightarrow{a} = \langle a_1, a_2, a_3 \rangle$ and $\overrightarrow{b} = \langle b_1, b_2, b_3 \rangle$, then the cross product of $\overrightarrow{a}$ and $\overrightarrow{b}$ is the ...
0
votes
1answer
26 views

Finding the gradient of a function.

A function $f=f(x,y)$ has continuous partial derivatives , and assume that maximal directional derivative of $f$ at $(0,0)$ is equal to $100$ and is attained in the direction towards $(3,-4)$ , we ...
2
votes
1answer
23 views

Jacobi determinant for high-dimensional sphere inversion

I need to find the Jacobi determinant for the unit sphere inversion in $\mathbb{R}^n$, i.e. the map given by $f(x) = \frac x {|x|^2}$ for $x\in \mathbb{R}^n$. The main problem is to figure out the ...
0
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0answers
20 views

How to find Tangent Normal Binornmal vectors of parametric knot

I am given parametric equations of torus knot: $$x = (a+b\cos(qt))\cos(pt)$$ $$y= (a+b\cos(qt))\sin(pt)$$ $$z= c\sin(qt)$$ where $0<t<2\pi$. I need to find Tangent, Normal and Binormal ...
1
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1answer
15 views

Finding the directional derivative.

We need to find the directional derivative of the function , $f(x,y) = x^{2}+y^{2}+xy$ at $P(1,-1)$ in the direction towards origin. The direction towards origin form the point $(1,-1)$ is ...
2
votes
2answers
24 views

Finding $P$ knowing $\overrightarrow{PQ}×\overrightarrow{b}$, $\overrightarrow{PQ}⋅\overrightarrow{c}$, $\overrightarrow{b}$, and $\overrightarrow{c}$

Let $Q$ be the point $(1,2,3)$, let $\overrightarrow{b} = \langle -1, 0, 1\rangle$, and let $\overrightarrow{c} = \langle 2, 1, 5\rangle$. It is known that $\overrightarrow{PQ} \times ...
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0answers
33 views
0
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1answer
12 views

Find $\overrightarrow{b}$ knowing $comp_{\overrightarrow{a}}\overrightarrow{b} = 5$ and $\overrightarrow{a}$

I am trying to solve the following question: Given that $\overrightarrow{a} = \langle3, 2, -1\rangle$, find a vector $\overrightarrow{b}$ such that $comp_{\overrightarrow{a}}\overrightarrow{b} = ...
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0answers
34 views

Inverse Function Theorem: is this true?

The inverse function theorem is usually stated as follows: Let $f:\mathbb R^n \to \mathbb R^n$ be a smooth map and let $x_0$ be a point such that $\det J_f (x_0) \neq 0$. Then there exists an open ...
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1answer
18 views

Obtaining z-transform of multivariate nonlinear difference equations

I need to obtain the z-transform of difference equations that are as follows: My problem however is multivariate and looks like this: $$ \begin{align} x_{k+1}&=ay_{k}+ x_{k}^2y_{k}\tag{1} \\ ...
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0answers
8 views

To prove that there is no scalar field $f'(a;y)>0$ for a fixed vector $a$ and every nonzero vector $y$.

To prove that there is no scalar field $f'(a;y)>0$ for a fixed vector $a$ and every nonzero vector $y$. I am having difficulty in this problem please help. Here $f'(a;y)$ is the derivative of $f$ ...
0
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3answers
49 views

Consider $F(x,y)=f(x+3y,2x-y)$…

If $f: \mathbb{R}^2\rightarrow\mathbb{R}$ where $F(x,y)=f(x+3y,2x-y)$ with $f$ is defferentiable and $\nabla f(0,0)=(4,-3)$ compute the derivate at the origin in the direction of unit vector ...
2
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1answer
22 views

Definition of differentiability at the point in multivariable calculus.

I'm self-studying the analysis from Zorich and the next definition of differentiability is given: $f:E\to \mathbb{R}^n$ is differentiable at the point $x$, which is a limit point of $E\subset ...
0
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0answers
36 views

Continuity at $(0,0)$ of $f(x,y)=2xy^2/(x^2+y^4)$ along the paths $φ(t)=(t,t)$ and $ψ(t)=(t^2,t)$

Let $f: \mathbb R^2→\mathbb R$, $φ: \mathbb R→\mathbb R^2$, $ψ: \mathbb R→ \mathbb R^2$ be given by $φ(t)=(t,t)$, $ψ(t)=(t^2,t)$, $t ∈ \mathbb R$ and $$f(x,y) = \begin{matrix} \frac{2xy^2}{x^2+y^4} ...
1
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1answer
36 views

Partial derivatives with component $e^{-y}$

I have to solve the equation $$f(x,y) = x^2e^{-y}$$ calculating the second partial derivatives for $x$ and $y$. I had no problem for variable $y$, I did it and it is correct. For the variable $x$ I ...
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0answers
46 views

For the classical diffusion equation ut = r (5ru) (in 3 space dimensions)

fi nd TWO changes of variables which changes the di ffusion constant from 5 to D = 1 for the new coordinate system?
1
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1answer
23 views

If $\nabla f(x,y,z) $ is always parallel to $xi+yj+zk$, them $f$ must be equal values at the points $(0,0,a)$ and $(0,0,-a)$.

If $\nabla f(x,y,z) $ is always parallel to $xi+yj+zk$, them $f$ must be equal values at the points $(0,0,a)$ and $(0,0,-a)$. I am having difficulty in the problem. Please help.
0
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1answer
46 views

Difficulty solving equation with partial derivatives

So when I was in school I never went past college algebra. But I have encountered a specific equation which I want to understand. At first I thought that an afternoon's focus would be enough to ...
4
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1answer
44 views

Splitting up a double integral

I need to compute the following integral: $$ 2\pi\nu^2\int^a_be^{x^2}\int_{-\infty}^xerfcx(-y)dydx, $$ where $erfcx(x)=e^{x^2}erfc(x)$, $erfc(x)=1 - erf(x)$, and $erf(x)$ is the error function. The ...
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0answers
20 views

Different Partial Derivatives Used in Curl in Proof of Stokes' Theorem

I was reading this proof of Stokes' theorem, which uses two different forms of partial derivatives in its syntax. The function at hand is $P(x, y, f(x, y))$, describing an arbitrary surface in ...
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2answers
49 views

Orthonomal bases and cross products

I want to show that if I have an orthonormal basis of $\mathbb{R}^3$, say $\{\boldsymbol{u}, \boldsymbol{v}, \boldsymbol{w}\}$, and if $\boldsymbol{u} × \boldsymbol{v} = \boldsymbol{w}$, then we have ...
3
votes
4answers
90 views

How I can evaluate $\lim_{(x,y) \rightarrow (0,0)} xy(\frac{1+xy}{x^3+y^3})^{1/3}$

I don't have idea how I can evaluate this double limit $$\lim_{(x,y) \rightarrow (0,0)} xy \left(\frac{1+xy}{x^3+y^3} \right) ^{1/3}$$ could you help me please! I try prove that $f$ is continuous: ...
0
votes
3answers
71 views

Limit $\lim_{\left(x,y\right) \rightarrow \left(0,0\right)} \frac{x^3+y^3}{\sin x^2+y^2}$ [on hold]

Find the limit of: $$\lim_{\left(x,y\right) \rightarrow \left(0,0\right)} \frac{x^3+y^3}{\left(\sin x^2 \right)+y^2}$$ How to find this limit? What is the most straightforward method?
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votes
2answers
31 views

On the Spivak's proof of the theorem 3-11 (calculus on manifolds)

In second paragraph of the case 1 within the proof: What is $U$ s.t $A\subset U$ and satisfies in the proof of the case 1 of theorem 3-11. $\psi_i$ is defined on $U_i$ and its support is not ...
-1
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0answers
23 views

Unable to perform cable theory equation [on hold]

First time posting, let me know if I'm in the wrong board. In my spare time I enjoy reading. I am currently reading Bioelectromagnetism. I stumbled across the cable theory equation for ...
0
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0answers
16 views

How to know function is increasing function for multi variable case?

If there is no leading power from the multi variable function, and if one of the partial derivative is negative and other partial derivative is positive How do I figure out whether the function ...
3
votes
2answers
91 views

Perturb a piecewise-linear path to make it $C^\infty$

I'm trying to prove that any two points on a path connected smooth manifold can be joined by a smooth path. It becomes easy if I can prove the following: Given a curve $\gamma :\mathbb{R} \to ...
1
vote
1answer
27 views

Calculate the flux through a closed surface

While studying for a test I have encountered such a task: Calculate the flux through a closed surface, where $S$ is a boundary of area $V$ with an outward orientation. The data: ...