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
34 views

How to show that $D \det_A (H)$ exists and equals $\det( adj(A)H)$?

Consider the function $\det : M_n(\mathbb R) \to \mathbb R$ ; how to show that for any $A , H \in M_n(\mathbb R)$ , the derivative operator of determinat of $A$ evaluated at $H$ i.e. $D \det_A (H)$ ...
1
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1answer
52 views

Defining a function that can take one OR two arguments.

This is a two part question: 1) Let's define a recursive function as so: $$f(x,y)= \begin{cases} \hfill f(x,5) \hfill & y\le0 \\ \hfill 0 \hfill & y=1\\ \hfill x+f(x,y-1) \hfill & y>...
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1answer
85 views

Gradient of function of matrix exponential

Suppose I have a differentiable function $\phi: \mathbb{R}^{p\times p} \mapsto \mathbb{R}$ defined as $\phi(\exp(tA))$ where $t$ is a positive scalar and $A$ is a $p\times p$ real matrix. How can I ...
0
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1answer
33 views

Can there be different values of $y_p$ for one equation?

For example, consider following example: Solution given by book is this: I solved it using different approach(as shown in the pic below) & got different answer. Is my solution wrong or ...
0
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0answers
19 views

Characterize $|\nabla f|$ as minimal function which satisfies an upper gradient inequality

Let $f \in C^1( \mathbb R^n, \mathbb R) .$ Then one by chain rule has $$ (*)\qquad |f(g(1))-f(g(0))| \leq \int_0^1 |\nabla f|(g_t)|g'(t) |\ dt, \quad \forall g \in C^1([0,1],\mathbb R^n). $$ I have ...
1
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1answer
23 views

Are scalar/vector fields in multivariable calculus related to fields of vector spaces in linear algebra

In linear algebra, I have learned that vector spaces are defined over fields. I have to admit that I don't have any background in abstract algebra, so my knowledge of fields are limited to $\mathbb R, ...
2
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1answer
30 views

Finding the maximum on an inside an octahedron

Let $B$ be the closed domain in $\mathbb{R}^3$ defined by $|x_1|+|x_2|+|x_3|\leq 1$. Find the maximum of $F(x_1,x_2,x_3)=\sum_{i=1}^3x_i^2+\sum_{i=1}^3a_ix_i$ on $B$. Using Lagrange multiplier ...
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1answer
36 views

How to find stationary points of two-variable cubic [closed]

I need to differentiate this cubic function to get the stationary points: $$f(x,y) = x^3 + ax^2 + bxy^2 + cxy + dx + e,$$ where $a$, $b$, $c$, $d$ and $e$ are constants. How do I do this?
1
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1answer
52 views

Verification of a Diffeomorphism

Below is an exercise to prepare for an Analysis II Exam Let $f: \mathbb{R} \to \mathbb{R}$ be a function of Class $C^1$ such that $|f'(t)| \leq k < 1$ for all $t \in \mathbb{R}$. Show that the ...
3
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2answers
69 views

How to find extrema of $\sqrt{x_1^2 + x^2_2 + x^2_3}$ defined on $\{x \in \mathbb{R}^3 : x_1^2 + 2x^2_2 + 3x^2_3 < 1\}$

I have a function $g: U \to\mathbb{R}$ where $$U :=\{x \in \mathbb{R}^3 : x_1^2 + 2x^2_2 + 3x^2_3 < 1\}$$ and $$g(x) = \sqrt{x_1^2 + x^2_2 + x^2_3}$$ I would like to find out if g(x) has any ...
0
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0answers
22 views

Finding the surface of intersection of 2 cylinders

Let $R=\{(x,y,z): x^2+z^2\leq 1, y^2+z^2\leq 1\}$. Compute the area of its boundary $\partial R$. The formula is $\int_D\sqrt{1+z_x^2+z_y^2}dxdy$ and $z=\sqrt{1-x^2}$, (I think). But what should the ...
1
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3answers
91 views

Prove an improper double integral is convergent

I need to prove the following integral is convergent and find an upper bound $$\int_{0}^{\infty} \int_{0}^{\infty} \frac{1}{1+x^2+y^4} dx dy$$ I've tried integrating $\frac{1}{1+x^2+y^2} \lt \frac{1}{...
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0answers
87 views

Find the maximum volume of the pyramid bounded by the plane and the coordinate planes?

Surface $\sqrt{c}=\sqrt{x}+\sqrt{y}+\sqrt{z}$ , $(c>0)$ I found that at $(x_{0},y_{0},z_{0})$ a tangent plane to the surface is : $\frac{x-x_{0}}{2\sqrt{x_{0}}}+\frac{y-y_{0}}{2\sqrt{y_{0}}}+\...
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2answers
42 views

Let $F$ be a vector field in $\mathbb{R}^3$. If $F$ is divergence free, we may deform the surface. Why?

In working through a solution, I can across the following generalization about vector fields and the Divergence Theorem. Can someone furnish a standard proof of this or at least its intuition? Let ...
3
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2answers
211 views

Maximize $xy^2$ on the ellipse $x^2+4y^2=4$

I was using Lagrange multiplier, any steps gone wrong? $$f(x,y)=xy^2$$ $$c(x,y)=x^2+4y^2$$ Partial Derivatives $$\frac {\partial f}{\partial x} = y^2 $$ $$\frac {\partial f}{\partial y} = 2xy $$ ...
1
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0answers
73 views

Integral-Summation Problem (Mathematical Physics problem)

given, $ψ_{k}$ =$\sqrt{\frac{2}{q}}$ $\sin \frac{kπx}{q}$ we have, $g_{kj}$=$q\int^q_0 ψ_j\frac{\partial ψ_k}{\partial q} dx$ verify, $\sum_{k}g_{jk}g_{lk}=q^2\int^q_0 \frac{\...
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0answers
23 views

Approximation of a 2 variable function.

If the weight of an object that does not float in water is $x$ pounds in the air and its weight in water is $y$ pounds , then the specific gravity of the object is : $S= \dfrac{x}{x-y}$ For a certain ...
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1answer
52 views

Find the minimum value of $(x+y)$

Two positive numbers $x$ and $y$ vary in such a way that $\ 128x^2-16x^2y+1=0$ Find the minimum value of $(x+y)$. The answer is 35/4, how do I get the answer?
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0answers
31 views

Approximating a two variable function.

A cylindrical tank is 4 feet high and has on outer diameter of 2 feet. The walls of the tank are 0.2 inches thick. We need to approximate the volume of the interior of the tank assuming that the tank ...
3
votes
3answers
880 views

Differentiability of a two variable function $f(x,y)=\dfrac{1}{1+x-y}$

We're given the following function : $$f(x,y)=\dfrac{1}{1+x-y}$$ Now , how to prove that the given function is differentiable at $(0,0)$ ? I found out the partial derivatives as $f_x(0,0)=(-1)$ and ...
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2answers
53 views

To find the critical points of $f(x,y)=e^{-x}(x^{2}-5xy^{2}+4y^{4})$ [closed]

I am having hard time finding the critical points of $$f(x,y)=e^{-x}(x^{2}-5xy^{2}+4y^{4})$$, but i could not find. Can anyone help. Thanks EDIT When i substituted $x=\frac{16}{10}y^2$ in first ...
1
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2answers
89 views

Computing a double integral over a surface S, where S is the unit sphere,

$$ \int \int_S (x^2+y^2)d\sigma$$ Where S is the unit sphere centered at (0,0,0), and $\sigma$ is surface area. I arrived at the correct answer of $\large \frac{8\pi}{3}$, but I took an (educated?) ...
0
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1answer
84 views

Stuck at the derivation of divergence in Cartesian coordinates.

I'll get to the point immediately. The definition of divergence in a point (from my textbook): $$ div \bar{E} = \lim_{V \to 0} \frac{1}{V}\oint_S \bar{E}.d\bar{S}$$ (it's a surface integral) ...
0
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0answers
17 views

What is the difference between ANOVA and ANOVA decomposition?

I was reading the paper about ANOVA decomposition http://faculty.bscb.cornell.edu/~hooker/fame_jcgs.pdf but I can't see how it is related to ANOVA. (Except that we have mutual orthogonality between ...
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2answers
59 views

How to interpret $\sum_{n\in \mathbb N^{d}} \frac{1}{n^{p}};$ and when it is converges?

I know that: $\sum_{n\in \mathbb N} \frac{1 }{n^{p}}$ converges if $p>1$ and diverges if $p\leq 1$ My Question is: What is an analogue this in more than one variable (say $d$)? Does it make ...
3
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3answers
286 views

One Step Forward from Gaussian Integral

Now to solve the integral $ \int_0^\infty e^{-x^2} \, dx $ has become a simple task for us. But how can we solve this integral: $$\int_0^\infty e^{-x^3} \, dx $$
4
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2answers
110 views

Apparent discrepancy between change of variables in one versus multiple dimensions.

My freshman calculus book gives the change of variables formula in one dimension and then eight chapters later gives it in $n$ dimensions. But when it generalizes to $n$ dimensions it requires the ...
3
votes
3answers
168 views

Hessian Matrix of an Angle in Terms of the Vertices

I am attempting to derive the analytical formula for the Hessian matrix of a the second derivatives of the value of an angle in terms of the (9) coordinates of the 3 3D points that define it. While I ...
3
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0answers
177 views

Rudin's Rank theorem

Rudin states the following: 9.32 Theorem: Suppose $m,n,$ are nonnegative integers, $m\geq r,n\geq r$, $F$ is a $C^1$ mapping of an open set $E\subset R^n$ into $R^m$, and $F'(x)$ has rank $r$ for ...
1
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0answers
38 views

What happens to tangential gradient when flattening a surface

The tangential gradient $\nabla_\tau f$ associated to a surface $S$ is defined as the projection of a suitable extension $\nabla f$ to the tangent plane to that surface. It seems reasonable to think ...
1
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1answer
27 views

Tangent vectors and parametric curves

Consider the curve $C$ defined by $(x,y,z) = \bar{r}(t)$, where $$\bar{r}(t)=\langle t\sin t, t\cos t, t^2 \rangle~~; t \in \mathbb{R}^3$$ Show that $C$ lies on the paraboloid $z= x^2 + y^2$ ...
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votes
1answer
110 views

Find partial derivatives, given directional derivatives. [closed]

You are given that the directional derivatives of a function $f$, at the point $(a, b)$, in the direction of the two vectors $(1, 2)$ and $(−1, 1)$, are $2$ and $3$ respectively. Find the partial ...
2
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4answers
293 views

show that out of all triangles inscribed in a circle the one with maximum area is equilateral

show that out of all triangles inscribed in a circle the one with maximum area is equilateral How do i start. I have to use function of two variables Thanks
1
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1answer
55 views

How do i determine maximum or minimum at (1,1) of function $ f(x,y)=(x-y)^{4} + (y-1)^{4}$

How do i determine maximum or minimum at this point of function $$ f(x,y)=(x-y)^{4} + (y-1)^{4}$$ I am getting doubtful case at point (1,1). How do i furthure investigate whether it is point of ...
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0answers
55 views

Open Set in the Cartesian plane.

I'm trying to prove that the following set is an open set in $\mathbb{R}^2:$ $$A=\{(x_{1},x_{2})\in\mathbb{R}^{2}: x_{1}+x_{2}>1\}$$ with respect to norm $||x||_{1},||x||_{2},||x||_{\infty}.$ ...
1
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2answers
28 views

Use Implicit Differentiation to compute the partial derivate $\frac{\partial z}{\partial x}$ at (1,1)

The question I am working on is: The equation $xy+z^3x-2yz=0$ defines z as a function x,y around the point (1,1,1). Use Implicit Differentiation to compute the partial derivate $\frac{\partial z}{\...
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2answers
163 views

Prove $\{(x,y): x>0\}$ is connected

As an introduction to multivariable calculus, I'm given a small introduction to some topological terminology and definitions. As the title says, I have to prove that $\{(x,y): x>0\}$ is connected. ...
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1answer
39 views

Is there a text introducing “high order Fréchet derivative” well?

Let $X,Y$ be Banach spaces and $U$ be open in $X$. High-order Fréchet derivatives are defined inductively so that the n-th Fréchet-derivative of a function $F$ is $F^{(n)}:U\rightarrow L(X,L(X,....,L(...
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2answers
116 views

Condition for equality of mixed derivatives

It says that theorems 12.11 and 12.12 imply Theorem 12.13. But, don't we need some extra conditions? Like existence of $D_{r,r}f$ and $D_{k,k}f$? Here $f$ is a function from $\mathbb{R}^n$ to $\mathbb{...
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2answers
36 views

Finding a good variable substitution for a double intergral

I want to compute the following integral $$ \iint_D{(x-y) dxdy} $$ where D is the triangle contained within these points: (0,0), (-2,1) and (-1,3). The lines that connect the the points form the ...
0
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1answer
38 views

Prove $\lim_{(x,y)\to (1,-1)} f(x,y)=1$

Let $f : \mathbb R^2 \to \mathbb R$ be defined by $f(x,y)= 3x + 2y$. Prove that $$\lim_{(x,y)\to (1,-1)} f(x,y)=1$$ I know that I must prove that for every $\epsilon>0$ we can find a $\...
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0answers
175 views

Sufficient conditions for integration by parts in higher dimensions

If $\Omega\subset {\mathbb R}^n$ is a bounded open set with $C^1$ boundary and $\nu$ denotes the outward unit normal to $\partial \Omega$, then the following formula holds for every pair of $C^1$ ...
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2answers
34 views

Proof of $\frac{\partial f}{\partial\vec u}(x_0,y_0) = ∇f(x_0,y_0)\cdot \vec u $

In order to prove that: $$\frac{\partial f}{\partial\vec u}(x_0,y_0) = ∇ f(x_0,y_0)\cdot \vec u $$ my book defines: $$g(t) = f(x_0+at, y_0+bt)$$ then, by the chain rule: $$g'(0) = \frac{\partial ...
4
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0answers
82 views

gateaux derivative and frechet derivative

In calculus, we have the following equation $DF(x,y)=\partial F_xdx+\partial F_ydy$ if $F$ is differentiable. I think such equation still holds for frechet derivative, but not for gateaux derivative. ...
3
votes
1answer
42 views

Tangent planes to $2+x^2+y^2$ and that contains the $x$ axis

I need to find the tangent planes to $f(x,y) = 2+x^2+y^2$ and that contains the $x$ axis, so that's what I did: $$z = z_0 + \frac{\partial f(x_0,y_0)}{\partial x}(x-x_0)+\frac{\partial f(x_0,y_0)}{\...
2
votes
2answers
115 views

What is an example of Gâteaux differentiable but not Fréchet differentiable at a point in a finite-dimensional space?

Let $V,W$ be nonzero normed spaces over $\mathbb{K}$ such that $V$ is finite-dimensional. Let $E$ open in $\mathbb{K}$ and $p\in E$. Let $f:E\rightarrow W$ be Gâteaux-differentiable at $p$. Is $f$ ...
1
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0answers
72 views

Bounding Expected Value of a piecewise function

Let X and Y to be two independent random variables with known pdfs. Get a bound for the expected value of the following expression in terms of $E[X]$, $E[Y]$, VAR[X] and VAR[Y]: \begin{equation} g(X,...
1
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0answers
81 views

Finding slope $dy/dx$ of tangent line to a curve defined in polar coordinates

Problem: Let the curve $f$ be defined by $ r= e^{\theta}$. 1) Compute the slope $dy/dx$ of the tangent line to $f$. Then use your result to define a function $g(x,\theta)$ that is a tangent line to $...
3
votes
3answers
93 views

Finding the distance from the origin to the surface $xy^2 z^4 = 32$ using the method of Lagrange Multipliers

Problem: Find the distance from the origin to the surface $xy^2z^4 = 32$. Attempt: The Lagrange equation for this problem is $L(x,y,z, \lambda) = x^2 + y^2 + z^2 + \lambda (xy^2 z^4 - 32)$. Setting ...
-1
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2answers
34 views

Lim of 2 variable function

Please help with this: Let $$f(z,b) = \begin{cases} z\sin(1/b), & b \ne 0, \\ 0, & b=0. \end{cases} $$ I'm trying to calculate the limit $f(z,b)$ for $(z,b)$ approach to $0$, in order to ...