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
15 views

Continuous differentiability of atan2

Consider the function atan2 defined on the plane, minus the origin and the negative $x$-axis, as the unique $\theta$ such that $-\pi<\theta<\pi$ and $$ x = r \cos \theta, \qquad y=r \sin ...
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2answers
17 views

Show that $\boldsymbol{\mathrm{F}}$ is independent of path.

Consider a vector field $\boldsymbol{\mathrm{F}}(x,y) = \langle 2xy, x^2 \rangle$ and three curves that start at $(1, 2)$ and end at $(3,2)$. Explain why ...
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0answers
9 views

Is the potential of a periodic conservative field periodic?

Let $Y = [0,1]^3$ and consider a conservative vector field $F$. Denote its scalar potential by $\varphi$, i.e. $$ \nabla \varphi = F. $$ If $\varphi$ is $Y$-periodic it is clear that $F$ is periodic, ...
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1answer
19 views

What are the minimum and maximum of $x$ and $y$ within the set $ 0 \le x \le 2$, $x - 2 \le y \le x$?

Given a set, how do I calculate what it's minimum and maximum is for x and y? $$ 0 \le x \le 2 \ , \ x - 2 \le y \le x$$ I informally look at it and think "if x is 0, then y is at most 0, and least ...
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0answers
18 views

Construction of a function

Give an example of a function that is partial differentiable and differentiable but not continuous partial differentiable . One example I thought (but is wrong) is the function: $$f(x, ...
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1answer
7 views

Tangent plane for level curve

Given $$f(x,y,z):\frac{x+2y+4xy}{5z^2 + 3}$$ what's the level curve going through the point $$p(6,1,-1)$$ and what's the tangent plane at that point to that level curve? What I've done is I've ...
0
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0answers
13 views

Differentiability find a and b

If I have the function $f(x,y)=|x|^a|y|^b$. How can I find the values of $a$ and $b$ that make the function differentiable on $\mathbb{R^2}$? How would I approach this problem I am quite confused?
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0answers
16 views

Gradient and hessian of square of quadratic form

I'm trying to differentiate a term of the form $(x^TA x)^2$ where $x$ is a vector and $A$ is a symmetric square matrix. Can anyone please tell me what the gradient and Hessian matrix of this term ...
1
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1answer
33 views

How to solve system of Differential Equations with 1 independent and 3 dependent variables

How can one solve this set of three differential equations in one independent variable "t" and three dependent variables A, B and F, which are functions of only t? $$ \frac{F(t) B''(t)+B'(t) ...
1
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1answer
22 views

Finding the points at which a surface has horizontal tangent planes

Find the points at which the surface $$ x^2 +2y^2+z^2 -2x -2z -2 = 0 $$ has horizontal tangent planes. Find the equation of these tangent planes. I found that $$ \nabla f = (2x-2,4y) $$ I'm thinking ...
1
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1answer
24 views

Is the line through $(-4, -6, 1)$ and $(-2, 0, -3)$ parallel to the line through $(10, 18, 4)$ and $(5, 3, 14)$?

Problem statement: Is the line through $(-4, -6, 1)$ and $(-2, 0, -3)$ parallel to the line through $(10, 18, 4)$ and $(5, 3, 14)$? My attempt: For the first line, we know the vector equation ...
1
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1answer
21 views

Multivariate Calculus - Partial Derivatives - Implicit Differentiation - Chain Rule

Let $z = z(x,y)$ be defined implicitly by $F(x, y, z(x,y)) = 0$, where $F$ is a given function of three variables. Prove that if $z(x,y)$ and $F$ are differentiable, then $$\frac{dz}{dx} = - ...
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1answer
22 views

Chain rule for $\ln(-f(\pmb{x}))$

I am trying to figure out how to calculate, for a smooth function $f:\mathbb{R}^{n}\rightarrow \mathbb{R}$ the second and higher order derivatives of $\ln(-f(\pmb{x}))$. I am not sure the notation, ...
0
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1answer
14 views

Find a vector equation and parametric equations for the line which passes through $(1, 0,6)$ and perpendicular to $x+3y+z=5$.

Statement of the problem: Find a vector equation and parametric equations for the line which passes through $(1, 0,6)$ and perpendicular to $x+3y+z=5$. I've gone through Calc. I, II in the ...
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0answers
17 views

Differences between directional derivatives

In our Calc 3 class, we have started doing directional derivative and their applications. So, for a function $f(x,y)$, the value of $f_{xx}f_{yy}-f_{xy}f_{yx}$ is used to determine what type of ...
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0answers
9 views

Multivariate Regression

Suppose there are $n$ variables that map through a function to a single output variable $r$. Given a set of 50-100 data sets with accepted input and output values that satisfy this relation, is it ...
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0answers
38 views

How do I do this change of variables?

Use a change of variables to evaluate: $$\iiint\limits_{D}xy\,\mathrm{d}V$$$D$ is bounded by the planes $y-x=0$, $y-x = 2$, $z-y = 0$, $z-y = 1$, $z=0$, $z=3$. I set $$u = y-x$$ $$v = z-y$$ $$w ...
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0answers
15 views

Stokes theorem problem [on hold]

Let $S$ be the torch-shaped surface defined by the cone $z = 1 + \sqrt{x^2 + y^2} $ for $2 \leq z \leq 4$, the cylinder $x^2 + y^2 = 1$ for $0 \leq z \leq 2$, and the disk $x^2 + y^2 ≤ 1, z = 0$. ...
1
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1answer
21 views

Can we do Taylor approximation in one direction

Let $f:\mathbb{R}^2\to\mathbb{R}$. Can we do Taylor approximation for only one variable $$f(x,y) \approx f(x_0,y) + \frac{\partial }{\partial x}f(x_0,y)(x-x_0) + \frac{1}{2}\frac{\partial^2}{\partial ...
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0answers
35 views

Proof of taking derivatives of both equal sides

I am curious about the proof of the following or whether the statement is true in general Assume that I have the following property: $f(x,y)=g(x,y,z)$ Can I assert that $D_xf=D_xg$ at any point ...
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0answers
14 views

Setting up a volume-finding calculation

I'm asked to find the volume inside the sphere $x^2+y^2+z^2=25$ and outside the cylinder $x^2+y^2=1$. I approached the volume $V$ in the following way: ...
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0answers
21 views

Implicit function theoroem [on hold]

Let $F = F(x, y, z)$ be a $C^1$ function which can be solved for any of the three variables in terms of the other two. Show that then $$ \frac{dx\,dy\,dz}{dy\,dz\,dx}=-1$$
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1answer
11 views

Non-singular derivative definition

I have a basic definition question. I am studying inverse function theorem, and I am stuck with what it means to say that for a $f'$ is non-singular? I looked it up in the internet, but it did not ...
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0answers
16 views

Partial Derivatives of a given function in $C^1$ [on hold]

Let $f:\Bbb R^3 \to \Bbb R$ be a given function in $C^1$. Find $d_xw$ and $d_yw$ in terms of $d_1f,\ d_2f,$ and $d_3f$ if $$w(x,y)=f(e^{x-3y},\ \sqrt{1+y^2},\ \ln(1+y^6))$$
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0answers
16 views

how to show the concavity of a very complex function?

To prove that S(n) have local maximum, I am thinking of taking second-order derivatives to $n$, then discussing the other parameter values. But S(n) seems way too complicated to me, I was wondering if ...
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0answers
18 views

Triple integral containing definite integral and exponentials with trigonometric functions

I am attempting to solve the following integral analytically: $$ \int_{z=5i}^{z=1} \int_{t=\csc^{-12}(z)}^{t=2} \int_{\theta=\sin^{t}(z)}^{\theta=t^2} {[\mathrm{e}^{t\cos(\mathrm{e}^{i \theta})} + ...
1
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1answer
35 views

Equivalent form of a double integral.

I am looking at the second question of this problem set: The iterated integral $\int_0^1 \int_{y/2}^1 e^{x^2} dx \, dy$ can be expressed as (a) $\int_0^1 \int_0^{2x} e^{x^2} dy \, dx$ ...
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1answer
13 views

Relation between minimum of a function and minimum of the sum of the same function and a linear term

I'd like to know if it's true that if given a function $f(x):X \mapsto \mathbb{R}$ and a vector $c \in X$, then if $$v = \arg\min_x f(x) + x^tc$$ one can say that $$v-c = \arg\min_x f(x)$$ Does this ...
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0answers
7 views

Looking for an alternative solution for optimal control problem

Let's say we have the following function ; $\intop_{0}^{\infty}\int_{0}^{N}V\left(C(t,\tau\right)dtd\tau$ and we want to maximise it according to the following constraint ; ...
1
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1answer
42 views

Prove whether a particular function is concave

Given the following equation: $$V(w) = - \frac{\alpha}{2} \left[ y_1(w) + y_2(w) + \int _{-\infty}^{+\infty} \vert y_1(w) - y_2(w) - x\vert f_{T1}(x)dx\right] \\- \beta \int _{w - y_1(w)} ^{+\infty} ...
0
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1answer
33 views

Evluating triple integrals via Spherical coordinates

Use Spherical coordinates to evaluate the triple integral $$\iiint_{\mathrm{x^2+y^2+z^2<z}}\sqrt{x^{2}+y^{2}+z^{2}}\, dV,$$ What I tried Converting $x^2+y^2+z^2<z$ to Spherical coordinates ...
2
votes
4answers
130 views

Find absolute maximum and minimum with domain

Find absolute maximum and minimum of the function $f(x,y)=3-x^2+y^2$ on the region $R = \{(x,y):1≥x≥0, 2≥y≥0\}$ I found that the gradient is $∇f(x,y)=(2x,2y)$ and that the critical point inside ...
0
votes
1answer
24 views

Differentiating both sides of an equality with respect to first variables? (Not answered)

I am proving a statement and the truth of the following proposition would help me with it. If anyone could say whether the proposition is true and give a hint how to prove it I would be very much ...
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0answers
41 views

A Riemann Integrability Question

Define $f:\mathbb{R} \rightarrow \mathbb{R}$. For any fixed closed interval $[a,b] $,$f(x) $ is $Riemann$ integrable on $[a,b].$ Proof:$\forall a,b;c,d\in\mathbb{R},a<b,c<d.$ $f (x+y) $ is ...
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2answers
42 views

How do I approach this double integral?

Let $R$ be the region inside $$x^2+y^2 = 1$$ but outside $$x^2+y^2 = 2y$$ with $x \ge 0 $ and $y \ge 0$ Let $$u = x^2 + y^2$$ and $$v = x^2+ y^2 - 2y$$ Compute $ \iint_R xe^y dxdy$ using this change ...
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0answers
24 views

2 variable limit

So, I understand why these bigger limit above does not exist (I'll name it 1), but I can understand why the other (2) is $0$. It seems to me that the $y^4/(x^6+y^8)$ is a non limited function and so ...
0
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0answers
17 views

Integration by parts partial derivatives

Given $$\int_x \int_t \Big( \frac{\partial}{\partial t}u(x,t) + \frac{\partial}{\partial x}f(u(x,t)) \Big) \phi(x,t)~~ dt dx = 0$$ How can I apply integration by parts in order to have the ...
1
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1answer
17 views

Rewriting triple integrals

I'm having trouble rewriting a triple integral. The question is rewrite the following integral in five different ways: $\int_0^1\int_y^1\int_0^z f(x,y,z) dx dz dy$ I am having trouble with ...
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2answers
53 views

Integration of $F(\sum_k x_k)$ over positive orthant

Problem Suppose we some function $F\left(\sum\limits_{k=1}^n x_k\right)$ over the positive orthant $[0,\infty)^n$. Show that this this is proportional to the integral $\int\limits_0^\infty ...
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0answers
9 views

Level curve of function with three variables

Sketch some level curves of the function f(x,y,z) = xy + yz, c=0 So when I set x=y=0, then z than can be any real number. Setting x=z=0, y can be any real number, same with x as well. Is this ...
2
votes
1answer
47 views

Compute a multiple integral$\iint_{[0,1]^2} (xy)^{xy} dxdy$

$$\text{Compute} :\iint_{[0,1]^2} (xy)^{xy} dxdy$$ I am thinking about changing the variable, $x=u,y={v \over u}$.But it doesn't work. I just found that the answer is$\int_0^1 t^t dt$.Maybe my idea ...
3
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2answers
16 views

Converting plane equation from $ax+by+cz=d$ to $r=a+\lambda b+\mu c$

The equation of the plane Π is $$2x + 3y + 4z= 48$$ Obtain a vector equation of Π in the form $r = a + λb + μc$, where a, b and c are of the form pi, qi + rj and si + tk respectively, and ...
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1answer
29 views

Change of variable in double integrals

I need help to solve the following question(s). a) Evaulate the integral $$\iint_D (x-y) \, dx \, dy,$$ where $D$ is the triangle with vertices $(0,0)$, $(-1,1)$ och $(4,2)$. b) Evaulate the ...
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0answers
35 views

How do I do this double integral (change of variable)

$B$ is the region bounded by $xy = 1$, $xy = 3$, $x^2 - y^2 = 1$, $x^2 - y^2 = 4$ Find $$\iint\limits_{B}x^2 + y^2 \,dx\,dy$$ using the change of variables: $$u = x^2 - y^2$$ $$v = xy$$ So I think ...
1
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0answers
26 views

What is the variance E[A]^2, statistics? [on hold]

$x(t)= A_i$, for $i \leq t < i + 1$ and $\{i = 0, 1 ,2 ,3,.....\}$. $A_i$ are independent variables, pmf of $P(A_i = 1) = P(A_i = -1) = 1/2$. Find the variance $E[A]^2$. I am so stuck on this ...
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votes
1answer
18 views

Intersection of Level Curves and a Ellipse at a given angle

I am preparing for an exam and I'm going over previous administered tests. I have come across the following problem and have little idea how to tackle it. It goes as follows: Let ...
2
votes
1answer
46 views

Inverting a function

I am stuck with the following problem I am supposed to find the inverse of the function $g$ with $2$ variables, where $$\begin{align*}g&: R^2\to R^2 \\ g&(x,y)=(2ye^{2x}, xe^y)\end{align*}$$ ...
0
votes
1answer
37 views

How to find the normal vector of $xyz=1$

How do I find the normal vector of $ xyz=1 $ at $(a, b, c)$? Is the answer below correct? Because some answers on here are saying that the normal vector is $$ \Delta F = (f_x,\:f_y,\:-1) $$ So ...
2
votes
0answers
29 views

Directional derivatives with given values.

At the point (1,2), the function f(x,y) has a derivative of 2 in the direction toward (2,2) and a derivative of -2 in the direction toward (1,1). Find f_x(1,2) and f_y(1,2). Find the derivative of f ...
0
votes
0answers
12 views

Divergence using suffix notation

I am trying to show the divergence of $v(r) = ∇r^n$ where $r =$ ||r|| using suffix notation. The solution I am given says: $∇ · v(r) = ∂^2_ir^n = n∂_i(r^{(n−2)}r^i) = n((n − 2)r^{n−4}r^2+i + ...