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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0answers
47 views

Finding the curvature from a set of datapoints

I have a set of 1. 1-d 2. 2-d data. I want to find the curvature at each single point. Till now I was using difference technique to find out the curvature, i.e, central difference at middle and ...
0
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0answers
32 views

Induction on derivatives

I have troubles understanding this induction proof: Let $$g(x) = \vert x \vert^{2k+1}$$ Show by induction: $$\frac{\partial ^N g(x)}{\partial x_{i_1} \dots \partial x_{i_N})} = cx_1n \dots x_iN \vert ...
4
votes
1answer
102 views

Condition for Continuity (two variable)

I came across the following question while studying for quals. This one is from a previous qualifier. I have a few ideas (which I'll mention below), but am stuck on how to complete the problem. Any ...
1
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2answers
56 views

A Point on Curve Where Tangent is Parallel

How do I find a point on a curve, $r=\langle 2,3t,5t^3\rangle$ in which tangent line is parallel to some plane. I thought to find the derivative first, but having trouble there as well. Anyway, this ...
0
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1answer
37 views

Local extrema in special directions

I am looking for the extrema of a function $G(y_1,y_2,y_3,y_4)$ subject to the constraint $y_1 = y_4 + y_2y_3.$ We know that $G$ is defined if $(y_2,y_3,y_4)$ is in the cylinder $\mathbb{D} \times ...
0
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1answer
62 views

Application of inverse function theorem?

I am not completely sure if this a direct consequence of the inverse function theorem. Assume that we have a function $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ that we can write in terms of ...
2
votes
1answer
43 views

derivative of a recursive vector-valued function

I have a recursive vector-valued function $$\mathbf{y}(t)=\mathbf{W}\mathbf{y}(t-1).$$ To compute the derivative of $\mathbf{y}(t)$ with respect to $\mathbf{W}$, do I need to use the product rule? ...
1
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1answer
59 views

Verify $\frac {\partial B} {\partial T} =$ $\frac{c}{(e^\frac{hf}{kT}-1)^2}\frac{hf}{kT^2}e^\frac{hf}{kT}$

Find an expression for $\frac {\partial B} {\partial T}$ applied to the Black-Body radiation law by Planck: $$B(f,T)=\frac{2hf^3}{c^2\left(e^\frac{hf}{kT}-1\right)}$$ The correct answer (I believe) ...
5
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2answers
68 views

If $\langle f'(x) \cdot v , v \rangle > 0$ then $f$ is injective

Question: Let $f: U \to \mathbb R^m$ differentiable at the convex set $U \subseteq \mathbb R^m$. If $$\langle f'(x) \cdot v , v \rangle > 0 , \,\,\, \forall\,\, x \in U, v \neq 0 \in \mathbb ...
1
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1answer
42 views

Coordinate calculation on a unit sphere

I'm writing a first person 3D game and I do not know the math behind what I need. I have 3 angles, a,b, and c. Angle a shows relation of x and z axis Angle b shows relation of y and z axis Angle c ...
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1answer
44 views

Orthogonal vectors and potential [closed]

given the potential $ψ(x;y)$, such that $dψ=−u_2dx+u_1dy$, why are $∇ψ=(−u_2;u_1)$ and $ψ(x;y)=c$ orthogonal vectors ? $c \in \mathbb{R}$ is a constant, and $\mathbf{u}(x; y) = (u_1(x;y); u_2(x;y))$, ...
0
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0answers
122 views

Cartesian product between manifolds

I was given the following exercise: Show that if $M$ is a $k$-manifold without boundary in $\mathbb{R}^m$, and if $N$ is an $l$-manifold in $\mathbb{R}^n$, then $M \times N$ is a $k+l$ manifold in ...
1
vote
1answer
42 views

Inverse of the complex exponential function, considered as a multivariable function

Consider the complex exponential function $g: \mathbb{C} \to \mathbb{C}, z \mapsto e^z$. When identifying $\mathbb{C}$ with $\mathbb{R}^2$ in the natural way, then $g$ can be considered as a ...
1
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2answers
60 views

Finding the vector (direction) on surface that has the minimal temperature (given by formula)

everybody! The temprature on a mountain described by: $T(x,y,z) = x^2 + y^2 + z^2$ The mountain desribed by $z = -x^2-y^2 +5$ A man, whos coordinates are $(1,1,3)$, wants to go on a direction, in ...
1
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1answer
15 views

Partial derivatives of all orders of linear map exist

If F is a linear map from R^n to R^m is it true that F is C^infinity, i.e. partial derivatives of all orders exist? My thought is that the answer should be "yes," because the derivative of F is just F ...
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0answers
25 views

Are these two compositions of two functions differentiable?

Assuming $U=\{x\in\mathbb{R}^2:x_1^2+x_2^2<1\}$ is the open unit circle in the plane and $f,g:U\rightarrow\mathbb{R}^2$ two functions with $f(0)=g(0)=0$. $f$ is Fréchet-differentiable in $0$, and ...
0
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1answer
90 views

Existence of Partials Imply the Existence of Gradient Vector?

Let $f$ be a scalar function of three variables. Then the gradient vector is defined by: I read here that the existence of partial derivatives at some point $(x_0, y_0, z_0)$ does not imply the ...
0
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0answers
32 views

How to get the potential and the gradient of this function?

How to calculate the potential $P_3:\mathbb{R}^3\rightarrow\mathbb{R}$ with the $\nabla P_3=f_3$ of the function $f_3:\mathbb{R}^3\rightarrow\mathbb{R}^3$ with $\large ...
3
votes
2answers
281 views

Curl: invariant under change of basis or not?

I wondered how the curl$$\text{rot}\mathbf{F}=\left( \begin{array}{ccc}\partial_y F_3-\partial_z F_2 \\ \partial_z F_1-\partial_x F_3 \\ \partial_x F_2-\partial_y F_1 \end{array} \right)$$of a vector ...
3
votes
1answer
24 views

finding polynomials to approximate a multivariable function

Let $U := B_1(0) \subseteq \mathbb{R}^2$, with $B_1(0) := \{(x, y) \in \mathbb{R}^2,\space \|(x, y)\| _1 < 1\}$. Now consider the function: $$g: U \to \mathbb{R}^2, (x, y) \mapsto ...
3
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1answer
38 views

diffeomorphism inbetween two subsets of $\mathbb{R}^2$

Consider the function $$f: \mathbb{R}^2 \to \mathbb{R}^2, \space\space f(x, y) := \pmatrix{x(1-y) \cr x y}$$ Now first, why is $f$ continuously differentiable? Then, I want to prove that $f$ ...
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0answers
30 views

Calculating the volume bounded by $z = 5$ and $z^2=x^2+y^2$ in 2 ways

I don't understand where is my mistake on calculating the volume by the second way. The volume that I want to calculate is bounded by $z = 5$ and $z^2=x^2+y^2$, so it is the upper part of the cone, ...
1
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2answers
62 views

second derivative of the composition of two multivariable functions

Let $U \subset \mathbb{R}^n$ be open, and let $\gamma: \mathbb{R} \to U$ and $f: U \to \mathbb{R}$ be to functions that are differentiable at least twice. I want to show that $\frac{d^2}{dt^2}(f ...
0
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2answers
32 views

Multivariable function as a set of functions

Consider a function $f:\mathbb{R}^n \to \mathbb{R}^m$. I've understood that it can be seen as: $f_i = (f_1,f_2,\ldots ,f_m)$, where $f_i: \mathbb{R}^n\to \mathbb{R}$. What are $f_i$ exactly? ...
0
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0answers
28 views

Multivariable-Calculus - an algorithm for this kind of questions?

The temprature on a mountain described by: $T(x,y,z) = x^2 + y^2 + z^2$ The mountain desribed by $z = -x^2-y^2 +5$ A man, whos coordinates are $(1,1,3)$, wants to go on a direction, in which he gets ...
0
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1answer
14 views

Finding the best direction for a bird escape from a radiation (function of 3 parametres)

I have this question. My bird is is in this point: (1,1,3) in 3D, and the source of the radiation is in that point too. What is the direction for her to fly from that point, if it wants to minimize ...
0
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1answer
22 views

Proving the image set of $g$ is open

Let $U\subset V$ be open in $\mathbb{R}^k$. Let $f:V \to \mathbb{R}$ be of class $C^r$. Define $g(\mathbf{x})=(\mathbf{x},f(\mathbf{x}))$, for all $\mathbf{x}$ in $V$. Is $g(U)$ necessarily open in ...
0
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2answers
54 views

Prove $f$ is Lipschitz on $K$

Let $f:\mathbb{R}^d\to \mathbb{R}$ such that it's partial derivatives are continuous. Let $K\subseteq \mathbb{R}^d$, a bounded set. Prove that $f$ is Lipschitz on $K$. My work: Since $f$'s ...
1
vote
1answer
25 views

Manifolds: show that this map is not a coordinate patch

Let $S^1$ be the subset of $\mathbb{R}^2$ given by {$(x,y)|x^2+y^2=1$}. We all know that $S^1$ is a 1-manifold in $\mathbb{R}^2$. I'm trying to prove that the following map:$$\alpha:[0,1) \to ...
0
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1answer
30 views

Stoke's Theorem Application on Cylinder

This is a question regarding Stoke's theorem's application. This is in regards to a problem from MIT OCW. My question is, referring to the answer provided, what closed surfaces are used in the proof ...
0
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1answer
63 views

Particular $f \in C_c^1$

is there a way to construct a function $f \in C_c^1( B(0,\frac{3R}{4}))$ such that $f|_{B(0,\frac{R}{2})}=1, \quad f|_{B(0,\frac{3R}{4})^C}=0$ and $0\le f\le 1$ everywhere such that $|Df(x)| \le ...
0
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0answers
21 views

Find all $t$ where coplanar

What are all real $t$ such that the vectors $a=<1,2,3>, b=<2,5,8>, c=<1,1,t>$ are coplanar. What I thought: I thought to find when $a\cdot (b \times c)=0$ However, the $t$ is ...
0
votes
1answer
176 views

Angle between diagonals of two faces on a cube

What is the angle between diagonals of two faces on a cube originating at the same vertex? What I have done: Vector representations of the diagonals joining the points $(0,0,0)$ to $(1,1,1)$ and ...
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0answers
24 views

Inverse functions from $\mathbb{R}^k$ to $\mathbb{R}^n$

Given the definition of a manifold, how do we define the inverse of the coordinate patch, or, more generally, of this kind of bijective functions?$$f:\mathbb{R}^k\to \mathbb{R}^n$$ It suffices to ...
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1answer
66 views

Comparing Two Multilinear Polynomials based on Multivariable Taylor Expansion

Given two linear functions $f(x)$ and $g(x)$ defined on real values, let's say that I want to show that $f(x) > g(x)$ for all real $x > t > 0$. According to the order-1 Taylor expansion at ...
6
votes
1answer
147 views

How to calculate the area of a region with a closed plane curve boundary?

Under the conditions of Green’s Theorem, the area of a region $R$ enclosed by a curve $C$ is $$\oint_C x \, dy=-\oint_C y \, dx=\frac{1}{2}\oint_C (x \, dy - y \, dx)$$ I tried to use the result to ...
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3answers
443 views

Volume of the region outside of a cylinder and inside a sphere

The cylinder is $x^2 +y^2 = 1$ and the sphere is $x^2 + y^2 + z^2 = 4$. I have to find the volume of the region outside the cylinder and inside the sphere. The triple spherical integral for this ...
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1answer
225 views

Point on surface where tangent plane is perpendicular to line.

I'm given the surface $ x^3-2y^2+z^2=27 $ and have to find where the tangent plane is perpendicular to the line described by \begin{align*} x &= 3t-5 \\ y &= 2t+7\\z&=1-t\sqrt2\end{align*} ...
0
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0answers
35 views

Proving multi-variable differentiability using the limit definition

I'm doing advanced calculus and I find it challenging to solve multi-variable limits while proving differentiability, more specifically 2 variable limits. could you show me how do I solve this limit?: ...
4
votes
3answers
108 views

How do one rigorously prove that the electric potential energy of an conducting sphere with charge $Q$ is $\frac{Q^2}{8\pi\epsilon_0R}$

How do one rigorously prove that the electric potential energy of an conducting sphere with charge $Q$ is $\frac{Q^2}{8\pi\epsilon_0R}$? Is integration the only way? Homogeneous charge distribution ...
1
vote
2answers
102 views

On proving the total differential.

I am following an open-course on multi variable calculus provided by MIT taught by Denis Auroux. The question I am about to ask is from this lecture. In the lecture Denis Arnoux gives a sketch proof ...
0
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1answer
53 views

Verifying Green's Theorem

If we have the line integral of F=$(x^2-2xy)dx+(y^2-x^3y)dy$ over a square with vertices at $(0,0)$ , $(2,0) ,(2,2) ,(0,2)$ I get the answer $24$ when doing the double integral in Green's theorem , ...
2
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1answer
218 views

Numerically find a potential field from gradient

I know that the gradient of a potential field/scalar field is a vector field, and given the function of the gradient I know how to integrate each component to get back the original scalar field. But ...
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3answers
284 views

Solve this functional equation:

Functional equations such as this one appear only once every several years on exams, so I feel it's hard to have a sure-fire way to approach the problem, unlike, say, solving a series convergence ...
0
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1answer
30 views

What is $\nabla\cdot A\nabla u$ for $A\in C^1(\mathbb{R}^n\to\mathbb{R}^{n\times n})$ and $u\in C^2(\mathbb{R}^n\to\mathbb{R})$?

Let $A\in C^1(\mathbb{R}^n\to\mathbb{R}^{n\times n})$ and $u\in C^2(\mathbb{R}^n\to\mathbb{R})$. How can we compute $\nabla\cdot A\nabla u$? I assume we need to apply some kind of product rule, but I ...
2
votes
1answer
30 views

What is $\nabla Au$ for $A:\mathbb{R}^n\to\mathbb{R}^{n\times n}$ and $u:\mathbb{R}^n\to\mathbb{R}$?

Let $A:\mathbb{R}^n\to\mathbb{R}^{n\times n}$ and $u:\mathbb{R}^n\to\mathbb{R}$. How can we compute $\nabla Au$? I assume we need to apply some kind of product rule, but I wasn't able to figure out ...
11
votes
1answer
169 views

Reducing multi-variable functions to a composition of 1- or 2-variable functions

There are some special functions of 3 or more complex variables that are analytic in some domain (a region in $\mathbb C^n$) with respect to each variable. To give some examples: the incomplete beta ...
0
votes
1answer
55 views

Manifold with boundary given as the pre-image of a subset of $\mathbb{H}^n$

Let $f \colon \mathbb{R}^{n+k} \to \mathbb{R}^n$ be a function of class $C^r$ for $r>1$. If $M = f^{-1}(0)$ and $0$ is a regular value of $f$, then we know (using implicity functions theorem) that ...
1
vote
1answer
36 views

How to see that this map is not smooth?

Let $f(x,y) = x^2$ and define $g(x,y) = {x \cdot f(x,y) \over x^2 + y^2 } = {x^3 \over x^2 + y^2}$. In this book here it is stated (on page 86) that this function does not extend to a smooth function ...
1
vote
1answer
118 views

Multivariable optimization- Nature of critical points when det of hessian matrix = 0

I'm struggling a bit with my multivariable optimization. Assuming the determinant of the hessian matrix ≠ 0 I have no issue, though when the det = 0 I get stumped. Example- $$f(x,y)=x^4+y^4-(x+y)^2$$ ...