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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4answers
38 views

How to use Cross Product Properites to do proof

How do I proceed with a proof for this question? Prove that: \begin{equation} (a \times b) \cdot (c \times d) = \begin{vmatrix} a \cdot c & b \cdot c \\ a \cdot d & b \cdot ...
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3answers
23 views

Problem about planes

Say we have $2x+3y+3z=0$ which is a plane. Does that plane have infinite dimensions (it is a 2D "object" — forgive me as I am not a mathematician — but each side has infinite length) or is it just the ...
2
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1answer
28 views

Example where partial derivatives commute but are not continuous.

I am looking for an example of a function $f:\mathbb R^2\to\mathbb R$ such that there is a point $x\in\mathbb R^2$ with the following properties: 1) All partial derivatives of second order exist in a ...
2
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2answers
21 views

Necessary condition for local maximum

Let $\Omega\subset \mathbb{R}^n$ open, bounded and let $f:\Omega\to\mathbb{R}$ be a $C^2$-function. I want to prove: Necessary for a interior maximum $x_0\in\Omega$ is that $D^2f(x_0)$ is negative ...
0
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1answer
22 views

Statement about the gradient

Let $f \in \mathcal C^1(\mathbb R^n).$ If there exists $u \in S^{n-1}$ such that $$\nabla f(x) \cdot u \geq 0 \quad\forall x\in \mathbb R^n,$$ then $f(u) \geq f(0)$. How to prove this statement?
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1answer
22 views

Applying chain rule on a quadratic form

I'm trying to apply the chain rule on a quadratic form: $\frac{dx^TAx}{dx}=\frac{dx^T(Ax)}{dx}=\frac{dx^T(Ax)}{dAx}\frac{dAx}{dx}=\frac{dx^T(Ax)}{dAx} A$ But I'm stuck here. I think ...
0
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0answers
19 views

Evaluating surface area

These are two exercises from Apostol calculus that I am struggling to set up the integrals correctly. The biggest problem for me is finding the correct region $T$ under the surface $S$. Compute the ...
1
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2answers
23 views

What is the area of the part of the surface $z=yx$ bounded by $x^2+y^2=1$?

A parametrization of the part of the surface $z=yx$ bounded by $x^2+y^2=1$ is \begin{align} x &= u \cos v \\ y &= u \sin v \\ z &= \frac12 u^2 \sin 2v, \end{align} or $$r(u,v)=u \cos v \, ...
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3answers
97 views

Books various maths subjects

I am a Civil Engineering student and i am planning on following physics in my career.I want to be ready for the advanced undergraduate courses that i will attend to,so i need to learn Differential ...
1
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1answer
13 views

Line integral of vector field/Why doesn't my solution work?

The question in its entirety: Determine for which constants A & B the vector field $$\mathbb{F} = (Axln(z))\mathbb{i} + (By^2z)\mathbb{j} + ((\frac{x^2}{z})+y^3)\mathbb{j}$$ is conservative. If ...
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0answers
18 views

How to solve this double integral problem? [on hold]

$$D: y \leq 1, x^2 \leq y$$ $$\iint_D (y+yxf(x^2+y^2))\,dx\,dy$$
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1answer
29 views

Show that $\|f\|^{2}$ attains a minimum value on the interior of $B$

I am looking for any help, hints, or suggestions in how to go about this problem from a previous qualifying exam. We are given a smooth mapping $f: U \rightarrow \mathbb{R}^{n}$ whose differential ...
1
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1answer
16 views

Need help finding tangent plane to a surface

I'm doing a Calc III homework problem, and I cannot seem to figure out what the correct solution is. $$ \text{Find the equation of the tangent plane to the surface }z = 9 y^{2} - 9 x^{2}\text{ at the ...
1
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2answers
103 views

Limits in multivariable functions

I tried to evaluate this limit but I can't see any limited function here (the limit exists). $\lim\limits_{(x,y)\to(0,0)}\frac{2x^2y}{x^4 + y^2}$ Thank you.
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0answers
22 views

Gradient and invariance under change of basis

I intuitively would be inclined to believe that the gradients $\nabla F_i$ of the components $F_1,\ldots,F_3$ of a vector field $\mathbf{F}:A\subset\mathbb{R}^3\to\mathbb{R}^3$, $\mathbf{F}\in ...
2
votes
1answer
19 views

Choosing a vector normal to a jordan curve that points “inside”

Let $\gamma=\partial K_1(0,0)$ be the circle with radius $r=1$ and origin $(0,0)$ in $\mathbb R^2$. Then for any $t_0$ we have $\gamma'(t_0)\neq \begin{pmatrix} 0 \\ 0\end{pmatrix}$. Let ...
0
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0answers
40 views

Is the unit square a $2$-manifold in $\mathbb{R}^2$?

I'm using the following definition of a (smooth) manifold: It's from J.Munkres "Analysis on Manifolds". This is an exercise taken from this book: Is the unit square $[0,1]\times [0,1]$ a ...
0
votes
0answers
16 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
16 views

Parameterization of an ellipsoid in spherical coordinates

\begin{align*} 25x^2+16y^2+z^2=1 \\ \frac{x^2}{4^2} + \frac{y^2}{5^2} + \frac{z^2}{20^2} = \frac{1}{20^2} \end{align*} The spherical coordinates are defined as, \begin{align*} x &= \rho ...
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0answers
31 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
67 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
26 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
votes
1answer
27 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
votes
1answer
38 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
31 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
40 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) ...
4
votes
2answers
43 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
35 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
35 views

Orthogonal vectors and potential [on hold]

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
28 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
votes
1answer
62 views

Proof inequality using Lagrange Multipliers

Is it possible: $a,b,c$ are non-negative real numbers for which holds that $a+b+c=3.$ Prove the following inequality: $$ 4\ge a^2b+b^2c+c^2a+abc $$ Is it possible using Lagrange Multipliers. I ...
1
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1answer
20 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 ...
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2answers
40 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
11 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
19 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
votes
1answer
30 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 ...
1
vote
1answer
40 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
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1answer
20 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
votes
1answer
29 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 $f$ ...
1
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0answers
29 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, ...
0
votes
2answers
26 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
votes
2answers
31 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? ...
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0answers
23 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
votes
1answer
11 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
votes
1answer
17 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
votes
2answers
43 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 ...
0
votes
1answer
13 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
votes
1answer
19 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
votes
1answer
53 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 ...