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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1answer
107 views

Which is the correct definition of stationary point for real-valued functions in Euclidean space?

Given a multivariable real-valued function $f$ whose first partials all exist (but which aren't all continuous) at $p$, it is possible that $f$ is not (totally) differentiable at $p$. But since the ...
0
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1answer
146 views

An strange triple integral I've never seen before

This is one of the integrals I have for homework, But I've never seen anything like this before, I don't know what to do with it. Does anyone know? $$\int_0^1 dz \int_z^1 dx \int_0^x e^{x^2} dy.$$ ...
3
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2answers
79 views

Expansion of $ \frac{1}{|\vec r -\vec r'|} $

I would like to show that: $$ \frac{1}{|\vec r -\vec r'|} =\frac{1}{r} + \frac{\vec r'\cdot r'}{r^3}+\frac{3 ((\vec r \cdot \vec r)^2 -\vec r^2 \vec r'^2 )}{2r^5} +\dots$$ What I derived so far is: ...
1
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2answers
69 views

Why does determining the nature of local extrema for $\mathbb R \to\mathbb R$ functions require twice continuous-differentiability?

In the text Elementary Classical Analysis, why does Marsden specify the condition "twice continuously differentiable" here? Isn't mere twice-differentiability sufficient for the purpose indicated? ...
0
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1answer
114 views

If a function is differentiable almost everywhere, can it be written as an integral?

Consider a function $f:\mathbb{R}^n \to \mathbb{R}$. If $f$ is differentiable with Lebesgue integrable derivative, we may write $$ f(x+y) - f(x) = \sum_{i=1}^p \int_0^1 y_i \nabla f_i(x+ty)dt $$ by ...
0
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2answers
105 views

Intuition behind $\vec{e_i} \times \vec{e_j}=\epsilon_{ijk} \vec{e_k}$ (Levi Civita)

Let $\vec{e_i}$ denote a unit vector. Then we can write: $\vec{e_i} \times \vec{e_j}=\epsilon_{ijk} \vec{e_k}$, where $\epsilon_{ijk}$ is the Levi Civita symbol. Can someone intuitively explain me ...
2
votes
1answer
136 views

Calculating $\Delta_r f(r) $; stuck with: $\nabla \cdot \hat{r}$

I would like to calculate $\Delta_r f(r) $. This is as far as I got: $\Delta_r f(r) =\nabla \cdot \nabla f(r) =\nabla \cdot \frac{\partial }{\partial r} f(r) \hat{r} = \nabla ( \frac{\partial ...
6
votes
1answer
218 views

How much can we “cheat” and use vector knowledge in complex analysis?

I'm an engineering-physics student taking a course in complex analysis, and it's a little frustrating, because I see all these connections to vector calculus over the reals (especially as applied to ...
0
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2answers
59 views

Non-elementwise Matrix Derivatives

Let A,B,C,D,X be matrices. I'd like to perform a Gradient Descent minimization to the loss functin $$ tr[(AXBX^TC-D)^T(AXBX^TC-D)] $$ My question is, how to take the gradient efficiently w.r.t. $B$? ...
0
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1answer
677 views

find the rate of change of a function at a point in the direction perpendicular to the a plane and moving away from the origin.

I am stumped on how to solve this type of problem. If anyone could give me a hand it would be appreciated. Let f(x,y,z)=x^2+y^2+z^2. At the point (1, 2, 1), find the rate of change of f in the ...
1
vote
1answer
535 views

Green's first and second identities

quick question on greens identities given that i know $\nabla \cdot (fv) = \nabla f \cdot v + f\nabla \cdot v$ then deduce $\int\int_\Omega \nabla f \cdot v$ $d{A} = \int_{\partial\Omega} fv\cdot ...
1
vote
2answers
44 views

Is the substitution given the question incorrect?

Show that an equation of the type $$u_{t}+6uu_{x}+u_{xxx}+a'(t)u_{x}=0$$ can be transformed into the KdV equation ($u_{t}+6uu_{x}+u_{xxx}=0$) by the transformation $$\begin{cases} \xi = x+a(t) \\ \tau ...
0
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1answer
40 views

Unclear on divergence of f*F

Let $f(x,y,z) = xyz^2$ and $F(x,y,z) = (xy, yz,zy)$. Compute divergince and curl of $G = (fF)$ Isn't that for $div(fF) = fdiv(F) + F\cdot \nabla f$? Or do I just multiply $f$ by $F$ to find $G$ and ...
2
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1answer
146 views

The symmetry of mixed partials, for derivatives of order > 2

Let $f\in C^r(A\subset \mathbb R^n,\mathbb R^m)$, $Df:A\subset\mathbb R^n\to L(\mathbb R^n,\mathbb R^m)$ so that $Df(x):\mathbb R^n\to\mathbb R^m$ is $f$'s total derivative, (abusing notation) ...
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0answers
20 views

A basic question on slope in a parametric curve

Consider a parametric curve $f(t) = (f_1(t),f_2(t),f_3(t))$ and consider two points on the curve (f_1(a),f_2(a),f_3(a)) and (f_1(b),f_2(b),f_3(b)). I want to know what vector is represented by the ...
1
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2answers
664 views

At what time is the speed minimum?

The position function of a particle is given by $r(t) = \langle-5t^2, -1t, t^2 + 1t\rangle$. At what time is the speed minimum?
0
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1answer
43 views

Multivariable Calc At what point does r(t) intersect the yz plane

The Position vector of a particle at time t is given by r(t) = <1/3(t^3 - 3t), t^2, 5> . At what point or points does the particle cross the yz-plane?
0
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1answer
138 views

Which is the correct Hessian matrix (the standard matrix of a bilinear form)?

Please note the typo in the first entry: $\frac {\partial^2f} {\partial x_1\partial x_2}$ should instead be $\frac{\partial^2f} {\partial x_1\partial x_1}$. Also, this Hessian matrix need not be ...
1
vote
3answers
213 views

How could you express the following double integral in term of a single integral?

How could I express the $$\int\int e^{(x^2+y^2)^2} dA$$ in terms of a single integral with respect to r where D is a disk with center (0,0) and radius 1
2
votes
1answer
29 views

Why am I getting half the correct answer by using Green's Theorem?

I had this homework problem that asked me to use Green's Theorem to solve it, so I did. Unfortunately, my answer was wrong. I looked for an error in my reasoning, but did not find it. I eventually ...
4
votes
2answers
72 views

Recovering vector-valued function from its Jacobian Matrix

Consider a function $f:\Omega\subset\mathbb{R}^n\rightarrow\mathbb{R}^m$, for which the Jacobian matrix $J_f(x_1,...,x_n)= \left( \begin{array}{ccc} \frac{\partial f_1}{\partial x_1} & ... ...
1
vote
1answer
27 views

Tangent vectors as curves equivalence relation

I do not understand the definition of the equivalence relation that is defined on the curves creating a tangent vector space. Let $X$ be any manifold, a point $x \in X$, two curves $\alpha:(-a,a) \to ...
0
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1answer
57 views

Change of variables with Partial Derivatives

I am doing a problem set and have come across the following question where I cannot get the right answer: A variable $z$ may be expressed either as a function of $(u,v)$ or of $(x,y)$, where $u = ...
1
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1answer
36 views

Does the transformation $x=Pabc$, $y=Qab(1-c)$, $z=Ra(1-b)$ map a unit cube to a tetrahedron?

Does the transformation $x=Pabc$, $y=Qab(1-c)$, $z=Ra(1-b)$ map a unit cube in $abc$ coordinates to the tetrahedron with vertices $(P,0,0)$, $(0,Q,0)$, $(0,0,R)$ and $(0,0,0)$ in xyz coordinates? ...
0
votes
1answer
55 views

What is the name of this map and what does it do?

I have found the following sentence and I am wondering about the meaning(what does this map do, as well as the name of this map g): Let $M:=f^{-1}(c)$ be an embedded manifold of dimension n-m with ...
2
votes
1answer
84 views

Setting up double integral — why not a triple integral?

The problem in question is: "Find the volume of the region bounded by $x^2+2y^2 = 2, z = 0$ and $x+y+2z = 2$." Am I setting up the integral correctly? This is equivalent to integrating the function ...
0
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2answers
597 views

If the Jacobian of two functions is zero, how are the two functions related?

Let $ x^{1} = f^{1}(u^{1},u^{2})$ and $x^{2} = f^{2}(u^{1},u^{2}) $. If the Jacobian of $f^{1}$ and $f^{2}$ is identically equal to zero (i.e. equal to 0 for all values of $u^1$ and $u^2$), why does ...
1
vote
0answers
32 views

$u\in C^1(\overline{\Omega}), v\in C^2(\overline{\Omega})$, then $F:=u\nabla v\in C^1(\overline{\Omega},\mathbb{R}^n)$?

Consider a limited $C^1$-domain $\Omega\subset\mathbb{R}^n$ and functions $u\in C^1(\overline{\Omega}), v\in C^2(\overline{\Omega})$. Is then $F:=u\nabla v\in ...
0
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1answer
39 views

Differentiabilty of this function

I want to show that $(x^2+y^2)^{\alpha}$ is not differentiable for $\alpha\in(0,1)$. All other cases are pretty straightfoward.
0
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1answer
45 views

How to obtain the Jacobian $J_{\dot{x}}(\dot{q})$ of the derivatives from the “normal” Jacobian $J_{{x}}({q})$?

I have a problem that arose in a kinematics context. Suppose, the Jacobian $J_{x}(q) = \frac{\partial{x}}{\partial{q}}$ of vector $x$ w.r.t. vector $q$ is known. I am interested in the Jacobian ...
0
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1answer
33 views

Allocation / Weighted Average question

I'm hoping someone can help with my problem. Forgive me as I don't even know what title to give this. I'm trying to come up with an allocation method for transportation expenses for multiple stops ...
0
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1answer
33 views

Derivative of a special function

I have the following formula, what is the derivative with respect to x? $$f(x) = \sum_{i=1}^{n} e^{x^Ty_i}$$
1
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1answer
39 views

why does (∂/∂t)(3fx+4tfy) = 3(∂/∂t)fx + 4fy + 4t(∂/∂t)fy?

z=f(x,y) , x= s^2+3t, y= s+2t^2 (∂z/∂t)= 3fx+4tfy why does (∂/∂t)(3fx+4tfy) = 3(∂/∂t)fx + 4fy + 4t(∂/∂t)fy ? I don't understand where 4fy comes from. Say it would have been (∂/∂t)(3t^2fx+4tfy) is ...
3
votes
2answers
58 views

Laplace's equation is solved when the functional $E[u] = \int_{\Omega}|\nabla u|^2 $ is minimized

My professor mentioned something like "Laplace's equation is solved when the functional $E[u] = \int_{\Omega}|\nabla u|^2 $ is minimized." I've been trying to understand this statement. If I say that ...
2
votes
4answers
103 views

How to prove that sequence $(1+1/n)^n$ is convergent and increasing?

For the sequence $(1+1/n)^n$, how does one prove that it is convergent and increasing series? I do know that as $n \to \infty$ it becomes constant $e$.
2
votes
1answer
145 views

Numerical methods for double integrals

Which methods are known to calculate double integrals like $$\displaystyle \int_{0}^{1}\int_{0}^{1} \frac {1}{x^y+y^x} dy dx$$ numerical ?
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2answers
107 views

Taking the cross product of a cross product? Proving an identity that involves gradients and vectors?

Problem 20: Solution: I am having difficulty understanding how the boxed is not equal to 0. The derivative of 1 is equal to 0.
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0answers
41 views

Is the Kelvin-Stokes theorem an “identity”?

From Wikipedia: An identity is an equality relation A = B, such that A and B contain some variables and give the same result when the variables are substituted by any values (usually numbers). ...
3
votes
1answer
68 views

Calculating a derivative using the chain rule vs. differentiating the composite

I have two functions $f:\mathbb{R} \rightarrow \mathbb{R}^2$ , $t \mapsto (t^3,t^2)$ and $g : \mathbb{R}^2 \rightarrow \mathbb{R}$ $(x,y) \mapsto (x^2+y^2)^{\alpha}$ Then we are asked to calculate $ ...
2
votes
1answer
110 views

Calculate double integral of …

I was doing a homework problem but now I'm stuck. The problem says: Calculate $\iint_{S} \frac{dx dy}{\sqrt{2a - x}}$ where S is a circle of radius $a$ which is tangent to to both coordinate axes and ...
0
votes
2answers
152 views

Volume of a triangular prism with non parallel bases

Consider an $\mathbf{(v_1,v_2,v_3)}$ triangle and its $\mathbf{\hat{n}}$ unit normal. Let $\mathbf{p_i}=\lambda_i\mathbf{\hat{n}} + \mathbf{v_i}$, $i=\overline{1,3}$. Is it possible to compute the ...
3
votes
5answers
68 views

Differentiability of Linear Maps

I am wondering whether all linear mappings have first-order partial derivatives (or stronger properties such as being continuously differentiable at all orders). Formally, suppose $A$ is an $m \times ...
1
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2answers
34 views

Multivaraible calculus,Global minima maxima

How do I compute the global minimum or maximum of the function $f(x,y)=-\sin x\cos y$. Given it is on a square $(0\leq x\leq 2\pi)$ and $(0\leq y\leq2\pi)$
4
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2answers
204 views

Taylor polynomial about the origin

Find the 3rd degree Taylor polynomial about the origin of $$f(x,y)=\sin (x)\ln(1+y)$$ So I used this formula to calculate it ...
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0answers
50 views

Implicit function theorem, system of equations

Problem statement Given the function: $$F(x,y,z) = (x^2y^3 + yz^3, xy^2 + y^3z^3) = (0,0)$$ Show that $F(x,y,z)$ is an implicit function $f: \mathbb{R} ^2 \to \mathbb{R}$, that is, $(x,y)^T = f(z)$, ...
1
vote
2answers
36 views

Evaluating a multivariable limit

$$\lim_{(x,y) \to (0,0)}\frac{3xy^2}{x^2+y^2}$$ No matter which way I approach this, I get 0. But how can I rearrange this equation into something that is defined?
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votes
1answer
31 views

Computing a formula for $\partial^2 f/\partial{v}\partial{w}$

Let $f(x,y,z)$ be of class $C^2$. Putting $x = u + v - w$, $y = 2u - 3v$, $z = v + 2w$ makes f into a function of u, v, and w. Compute a formula for $$\frac{\partial^2f}{\partial{v}\partial{w}}$$ in ...
0
votes
1answer
26 views

Equivalent parametric curves have the same length

I'm having trouble proving that length is preserved under equivalent curves. Equivalent curves are defined as such: Two non-closed piecewise smooth parametrized curves are equivalent $\phi : [a, ...
-1
votes
2answers
58 views

How to show a function is $1-1$ and onto?

Determine if the following function $T: \mathbb{R}^2\to\mathbb{R}^2$ is one to one and/or onto: $$T(x,y)=(x^2,y)$$
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3answers
80 views

When is $f_{xy}(x,y)\neq f_{yx}(x,y)?$

When is $f_{xy}(x,y)\neq f_{yx}(x,y)?$, where $f_{xy}$ and $f_{yx}$ denote the mixed (second) partial derivatives of a multivariable function $z=f(x,y)$.