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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-2
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0answers
48 views

$ \frac{\partial}{\partial x}(\frac{\partial f(x,y,z)}{\partial x}+\frac{\partial f(x,y,z)}{\partial y}\frac{\partial y(x,z)}{\partial x})=0$ [closed]

I need help, I dont understad how it do $ \frac{\partial}{\partial x}(\frac{\partial f(x,y,z)}{\partial x}+\frac{\partial f(x,y,z)}{\partial y}\frac{\partial y(x,z)}{\partial x})=0$ please please ...
2
votes
3answers
51 views

Proving that a set is open using epsilons.

I'm trying to prove that the set $$A=\{x=(x_{1},x_{2})\in\mathbb{R}^2:x_{1}^{2}+x_{2}^{2}>1\}$$ is open in $\mathbb{R}^2$ with the usual norm is open with the definition of "epsilons". My attempt ...
1
vote
0answers
20 views

What exactly is the critical point of ln(xy)

Can $\ln(xy)$ be a strictly concave function without a critical point? It seems that the graph has no critical point, therefore there doesn't seem to be maximum point. What does this imply?
0
votes
0answers
16 views

Understanding the meaning behind plotting a “gradient vector” on a graph containing contour lines?

A rather basic question here, please do forgive any technical errors in the question. Throughout this example consider the general function w=f(x,y) I am used to visualizing derivatives the same way ...
1
vote
1answer
22 views

Characterizing connected sets of $\mathbb R^n$ is terms of differentiable maps for which zero derivative everywhere implies constant

Let $U $ be an open subset of $ \mathbb R^n$ ; then how to prove that $U$ is connected iff for every differentiable function $f:U \to \mathbb R$ , $\nabla f(x)=0 \implies f $ is constant on $U$ ?
0
votes
3answers
26 views

Is the total differential the same as the directional derivative?

The way I understand it, the total differential and the directional derivative are both linear approximations of the change in a function at a certain point. So if I know the change in $x$ and $y$ ...
2
votes
1answer
29 views

System with arbitrary function of an unknown

How can I solve the following system $$ (u_x)^2 - (u_t)^2 = 1 \\ u_{xx} - u_{tt} = f(u) $$ where $f$ is an arbitrary function of $u$, $u$ and $f$ to be determined. I don't know any approach, ...
1
vote
2answers
20 views

How can I show that the limit of this function under these conditions does not exist?

Show that the limit of the function, $f(x,y)=\frac{xy^2}{x^2+y^4}$, does not exist when $(x,y) \to (0,0)$. I had attempted to prove this by approaching $(0, 0)$ from $y = mx$, assuming $m = -1$ and ...
1
vote
1answer
43 views

Proof of Green's identity

Can anyone explain to me how to prove Green's identity by integrating the divergence theorem? I don't understand how divergence, total derivative, and Laplace are related to each other. Why is this ...
0
votes
1answer
13 views

If points cannot be added, then how can we define $\lim_{m \to \infty}(a_m+b_m)$ where $a_m$ and $b_m$ are sequences of points in $\mathbb R^n$?

I am following Hubbard's multivariable calculus book. In the beginning of the book, it says that points cannot be added but vectors can. As a rule, it doesn't make sense to add points together, ...
2
votes
1answer
46 views

Derivative exists by first principles but undefined when using chain rule

Consider the function $h$ defined by \begin{align} h(z,y)=(z^3+y^3)^{\frac{1}{3}} \end{align} Then \begin{align*} h_z(0,0)&=\lim_{t\rightarrow 0}\frac{(t^3)^{\frac{1}{3}}}{t}\\ &=1 ...
4
votes
0answers
24 views

Why is n-th Fréchet derivative symmetric?

Let $V,W$ be nonzero normed spaces over $\mathbb{K}$. Let $E$ be open in $V$ and $f:E\rightarrow W$ be a twice Fréchet-differentiable function. Then, $D^2 f: E\rightarrow \mathscr{L}_2(V^2,W)$ is ...
0
votes
1answer
42 views

How is the Directional Derivative a linear transform?

So I know basically what a directional derivative is and how to calculate it using the gradient vector, but I'm a bit lost on the more advanced approach of looking at it as a linear transform. I've ...
2
votes
0answers
37 views

A simple question on the Lipschitz property

Suppose $f:\mathbb{R}^n\to\mathbb{R}$ is differentiable and $L$-Lipschitz, namely $$|f(x) - f(y)|\leq L ||x-y||_2 \ \ \forall x,y\in\mathbb{R}^n~.$$ How does this imply $$||\nabla f||_2\leq L\ ?$$ ...
0
votes
1answer
26 views

average height of a point on an arc vs hemisphere

Why isn't the average height of a point on an arc of radius a the same as the average height on a surface of radius a. Stated another way the first problem is: Find the average height of a point on a ...
2
votes
1answer
31 views

$f(x)$ be the characteristic polynomial of a matrix $A \in M_n(\mathbb R)$ ; then is it true that $f(1)=1+\operatorname{trace}(A)+O(\|A\|^2)$?

Let $f(x)$ be the characteristic polynomial of a matrix $A \in M_n(\mathbb R)$; then is it true that $f(1)=1+\operatorname{trace}(A)+O(\|A\|^2)$ ? I need a proof if it is true ; or any modification ...
1
vote
1answer
24 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
vote
1answer
37 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 & ...
0
votes
1answer
60 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
votes
1answer
30 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
votes
0answers
14 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
vote
1answer
20 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
votes
1answer
24 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 ...
-1
votes
1answer
25 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
vote
1answer
28 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
votes
2answers
58 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
votes
0answers
20 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
vote
3answers
73 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 ...
0
votes
0answers
23 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 : ...
1
vote
2answers
27 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
votes
2answers
106 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
vote
0answers
30 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 ...
0
votes
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 ...
0
votes
1answer
43 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?
1
vote
0answers
15 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
69 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 ...
-2
votes
2answers
49 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
vote
2answers
41 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
votes
1answer
64 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
votes
0answers
11 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 ...
1
vote
2answers
58 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 ...
2
votes
3answers
258 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: $$\int_0^\infty e^{-x^3} \, dx $$
2
votes
2answers
54 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 ...
2
votes
2answers
79 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 ...
2
votes
0answers
34 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
vote
0answers
14 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
vote
1answer
16 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 + ...
-2
votes
1answer
45 views

Find partial derivatives, given directional derivatives.

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
votes
4answers
69 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
vote
1answer
44 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 ...