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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44 views

Finding a perpendicular plane

Question Plane $B$ contains the points $(4, 2, 1)$ and $(4, 1, -6)$, and it is perpendicular to the plane $7x+9y+4z=18$ . What is the equation of the plane $B$? What I tried: I know that to find ...
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1answer
29 views

a question regarding gradients

I had a question in my exam, in which I had to practically find the gradient of a function in a certain point, thing is one of the vectors becomes 0 and I can't figure out how this happened, it says ...
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1answer
29 views

To show that the function $f:M_n(\mathbb R) \to M_n(\mathbb R)$ given by $f(A)=AA^t$ is differentiable and evaluate its derivative

How to show that the function $f:M_n(\mathbb R) \to M_n(\mathbb R)$ given by $f(A)=AA^t$ is differentiable and how to find the total differential at a point $X$ i.e. how to find $D f_A(X)$ ?
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1answer
24 views

To show that the map $f:M_n(\mathbb R) \times M_n(\mathbb R) \to M_n(\mathbb R)$ given by $f(A,B)=AB$ is differentiable and evaluate the derivative

How to show that the function $f:M_n(\mathbb R) \times M_n(\mathbb R) \to M_n(\mathbb R)$ given by $f(A,B)=AB$ is differentiable and how to find the total differential at a point $(X,Y)$ i.e. how to ...
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1answer
25 views

Definition of even functions for n dimensions

Is there a generalisation of even functions for functions with multiple variables? If so, what are some concrete examples of the use of this definition?
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1answer
86 views

Proof using the definition of limit that $\lim _{(x,y)\to(0,1)}\frac{x^2-y^2}{x^2+y^2} = -1$

Proof using the definition of limit that $$\lim _{(x,y)\to(0,1)}\frac{x^2-y^2}{x^2+y^2} = -1$$ and $$\lim_{ (x,y)\to(0,0)}\frac{ (1-\cos(xy))\sin y}{(x^2+y^2) }= 0$$ Definition of limit: $...
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1answer
34 views

$E(Y)$ from $f(x,y)$ where $0<x<y<∞$

A device contains two circuits. The second circuit is a backup for the first, so the second is used only when the first has failed. The device fails when and only when the second circuit fails. Let $X$...
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0answers
179 views

When the function is neither concave nor convex?

If I remember correctly multivariate function is convex when $f''_{11}(x,y) \leq 0, f''_{22}(x,y) \leq 0$ and $f''_{11}(x,y)f''_{22}(x,y) - (f''_{12}(x,y))^2 \geq 0$ and concave when $f''_{11}(x,y) ...
3
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2answers
293 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 ...
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1answer
43 views

Region enclosed by circle and line help

Need some help on a calculus assignment. Completely stumped and my long distance lecturers are non existent (as always) and my study guide doesn't have enough examples to be useful for this question. ...
2
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3answers
59 views

Why does $\left(\int_{-\infty}^{\infty}e^{-t^2} dt \right)^2= \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-(x^2 + y^2)}dx\,dy$?

Why does $$\left(\int_{-\infty}^{\infty}e^{-t^2}dt\right)^2 = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-(x^2 + y^2)}dx\,dy ?$$ This came up while studying Fourier analysis. What's the ...
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1answer
21 views

Expand the following expression : $\frac{1}{(x+\sqrt{(x^2+y^2)})}+\frac{x}{(\sqrt{(x^2+y^2)} (x+\sqrt{(x^2+y^2)}))}$

I have the following expression : $$\frac{1}{(x+\sqrt{(x^2+y^2)})}+\frac{x}{(\sqrt{(x^2+y^2)} (x+\sqrt{(x^2+y^2)}))}$$ I don't understand why : $$\frac{1}{(x+\sqrt{(x^2+y^2)})}+\frac{x}{(\sqrt{(x^2+...
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0answers
22 views

Another formula for second partial derivative of covariance function

So the problem arises from Stochastic Processes theory (particularly covariance function of the derivative of $L_2$-process) and is formulated like this: It can be proved that $$C_{X'}(s,t)=\lim_{(\...
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2answers
137 views

How do ideas in differential geometry expand upon ideas from introductory calculus

I just went through first year in mathematics and used Stewart's book for calculus. I am trying to self study differential manifold and I find many concepts such as chart, atlas very similar to that ...
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1answer
61 views

Generalisation of fundamental theorem of calculus (and density)

I wondered whether it holds, for a function $f:A\subset\mathbb{R}^3\to\mathbb{R}$, $f\in C(A)$, that, for all $(x_0,y_0,z_0)\in A$, $$\lim_{\sqrt{h_x^2+h_y^2+h_z^2}\to 0}\frac{1}{h_x h_y h_z}\int_{z_0}...
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1answer
82 views

Electric field due to a line of charge with non0uniform charge density

Question: The portion of the z-axis for which $-2<z<2$ m carries a nonuniform charge density of $z^2 + 1$ nC/m in free space. There are no other charges anywhere. Find $\vec{E}$ at a) $P_{A}(1,0,...
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3answers
5k views

Proof of the Gauss-Green Theorem

I can't seem to find any references that gives a proof of the Gauss-Green theorem: Let $U\subset\mathbb{R}^{n}$ be an open, bounded set with $\partial U$ being $C^1$. Suppose $u\in C^{1}(\bar{U})$,...
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0answers
43 views

Why is existence not guaranteed for this initial value problem using Existence and Uniqueness Theorem?

Given $\frac{dy}{dx}=\sqrt{x-y}$; $\ \ $ $y(2)=2$ Why is existence not guaranteed using the Existence and Uniqueness Theorem for Differential Equations? I thought that if $f(x,y)$ was continuous "...
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2answers
33 views

Find a tangent plane

I am asked to find a tangent plane of $f(x,y) = e^{x\ln y}$ at the point (2,1). When I ask wolfram alpha this, I am given the line $z=2y-1$. I don't intuatively understand this, shouldn't there be a ...
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1answer
75 views

Expression for the gradient using Taylor's Theorem

I've just started reading Nocedal and Wright's book on Numerical Optimization. On page 14 there is a formula for the value of the gradient in some point (equation 2.5) that I cannot derive myself. ...
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0answers
69 views

Gradient of an exponential function having a dot product of vectors

I want to show below relationship; $$\nabla_{j}\ e^{i\vec{p}.\vec{R}}= i\ p\ \hat{p_{j}}\ e^{i\vec{p}.\vec{R}}$$ where $\vec{p},\vec{R}$ have components in $j,k$ directions and $i=\sqrt(-1)$. I ...
2
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1answer
47 views

Integral over a torus - change of variables

Heyyyy I would like to understand how to perform the variables change $\phi \rightarrow \psi$ in the following sum $$ \int_{\varphi ...
2
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1answer
38 views

Simple questions about the line integral.

I learned very recently about line integrals in my class, and I've been given the definition: $$\int _\Gamma \vec {F} \cdot \vec {ds}=\int _a^b \vec F(\vec \alpha(t))||\alpha'(t)||dt$$ Where $\alpha$ ...
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1answer
33 views

Difficulty calculating velocity after lorentz transformation

I'm working on understanding Lorentz transformations via a text by Garrity, "Electricity and Magnetism for Mathematicians: A Guided Path from Maxwell's Equations to Yang-Mills". On pages 43 and 44 he ...
2
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1answer
26 views

Admits partials, definite integral of multivariable function is $C^1$.

For open $U \subseteq \mathbb{R}^n$, assume $f: [a, b] \times U \to \mathbb{R}$ admits partials $\partial_{x_i}f(t, x_1, \dots, x_n)$ continuous on $[a, b] \times U$. Do we have that $I_f(x_1, \dots, ...
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3answers
94 views

Showing that $\lim_{(x,y) \to (0, 0)}\frac{xy^2}{x^2+y^2} = 0$

Show that $$\lim_{(x,y) \to (0, 0)}\frac{xy^2}{x^2+y^2} = 0$$ I have tried switching to polar coordinates but I'm not getting a single term. This is what I did. Putting $$x=r\sin θ,\quad y=r\cos ...
2
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1answer
161 views

Using Partial Derivatives to find the equation of a tangent plane

$z = 4(x-1)^2 + 5(y+3)^2 +1$ at the point $(2,-2,10)$ I'm not sure how exactly how to proceed through the problem. I know to find the derivative with respect to $x$ , $y$, and $z$, which I did. For $...
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1answer
46 views

Alternative definition of differential

I've been thinking about idea of derivative, in particular about multivariable case. It seems that it is be pretty well-defined geometrically, so I thought maybe there is another definition, possibly ...
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0answers
20 views

Special and simplest coordinates for a vector field in a neighborhood of a point

If $v$ is a smooth vector field in a smooth manifold $M$, there always exist a chart over any $p \in M$ such that, in a neighborhood of $p$, $v$ has coordinates $v^1=1$ and $v^j = 0$, for all $j=2,...,...
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1answer
64 views

On the implicit function theorem and the gradient.

I was following some MIT notes and came across this proof I had a doubt about the existence of $r(t) = \langle x(t), y(t), z(t) \rangle$ a parametrization of a curve on the level surface. Then I ...
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1answer
37 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 3-space....
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3answers
261 views

Is there limit $ \lim_{(x,y) \to (0,0)} \frac{x^3}{x^2 + y^2}$?

How to show if the limit $$ \lim_{(x,y) \to (0,0)} \frac{x^3}{x^2 + y^2}$$ exists? I suspect that there is, as I can't find any path that would show that limit doesn't exist, and WolframAlpha also ...
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3answers
89 views

Help with proof of theorem 1-10 spivak calculus on manifolds

Theorem 1-10 The bounded function (mapping to $\mathbb{R}$), $f$ is continuous at $a$ if and only if the oscillation of a point of $f$ at $a$ is $0$ The definition spivak provides of oscillation ...
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0answers
30 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 ...
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1answer
47 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 ...
1
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1answer
25 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 ...
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0answers
16 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 g)(x_1,x_2)...
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1answer
41 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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1answer
33 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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0answers
59 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 ...
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2answers
58 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 need ...
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1answer
24 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 $$...
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1answer
115 views

Regarding the proof of $\int_0^xf(u)(x-u)du$

This question has already been asked: For continuous function $f$, prove: $\int_{0}^{x} \; \left[\int_{0}^{t}f(u) \;du \right] \;dt=\int_{0}^{x} f(u)(x-u)du$ but I really don´t understand the part ...
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1answer
43 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 ...
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1answer
58 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 $$\frac{\...
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2answers
52 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 ...
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1answer
37 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 ...
1
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1answer
24 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,\ x\...
1
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2answers
58 views

If the iteration $x^{k+1}=x^k-t_kH_k^{-1}\nabla f(x^k)$ converges superlinearly to a stationary point $x^*\ne x^k$, then $t_k\to 1$

Let $f\in C^2(\mathbb{R}^n)$ $(H_k)_{k\in\mathbb{N}_0}\subseteq\text{GL}_n(\mathbb{R})$ $x^0\in\mathbb{R}^n$ and $$x^{k+1}:=x^k+t_k d^k\;\;\;\text{for }k\in\mathbb{N}_0\tag{1}$$ with $(t_k)_{k\in\...
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0answers
75 views

Partial derivative of recursive exponential $f(x) = \sum^{K_2}_{k_2=1}c_{k_2} \exp(-z^{(2)}_{k2})$ with respect to the deepest parameter

I was trying to take the derivative of the following equation (which can be depicted nicely in a tree like structure, look at the end of question for diagram): $$f(x) = f([x_1, ..., x_{N_p}])= \sum^{...