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
30 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
27 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
34 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
27 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 ...
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0answers
36 views

Area under mapping

Can some one help me with this problem: Let $f:R^2\rightarrow R^2$ be defined by $\displaystyle{f(x,y)=(e^{x+y},e^{x-y})}$ Find the area of the image of the region $\{(x,y) \in R^2 : 0<x,y<1 ...
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1answer
46 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) ...
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1answer
1k 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 ...
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1answer
782 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
179 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
35 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, ...
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1answer
2k 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
30 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
68 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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1answer
51 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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1answer
90 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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1answer
57 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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0answers
16 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 ; ...
2
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1answer
58 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} ...
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1answer
59 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 ...
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4answers
431 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 ...
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1answer
35 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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2answers
76 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
30 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 ...
1
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1answer
48 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
88 views

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

Problem Suppose we integrate 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 ...
2
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1answer
61 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
30 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
59 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
53 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 ...
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1answer
110 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
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1answer
64 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 $$\begin{align*}g&: R^2\to R^2 \\ g&(x,y)=(2ye^{2x}, xe^y)\end{align*}$$ ...
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1answer
58 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
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1answer
43 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 ...
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2answers
57 views

Determining when $f(x,y) = x^{4/3} \sin(y/x)$ ($x \ne 0$) is differentiable.

Let $f$ be defined as follows: $$ f(x,y) = \left\{ \begin{array}{lr} x^{4/3} \sin(y/x) & \mathrm{if\ } x \ne 0 \\ 0 & \mathrm{if\ } x = 0. \end{array} \right. $$ I am asked to determine where ...
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1answer
100 views

Intermediate Integration Question

I'm having difficulty understanding why $$\int \left[ \left(\frac{dy}{dx}\right) ^2 + \left( y \right) \left( \frac{d^2 y}{dx^2} \right) \right]dx = \left( y \right) \left( \frac{dy}{dx} \right)$$
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1answer
142 views

Splitting partial derivatives

How come $$\frac{\partial f}{\partial x} = \frac{\partial f}{\partial u} \frac{\partial u}{\partial x} + \frac{\partial f}{\partial v} \frac{\partial v}{\partial x}$$ when $$ u = x\; cos \theta - ...
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2answers
383 views

Showing that a multivariable function is one to one

I am stuck with the following problem I am given the function $f$ such that $f(x,y)=(x^2-y^2,2xy)$ I am supposed to show that the function is one to one. For a function to be one to one, $f'>0$. ...
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2answers
218 views

Can a region always be parametrized by a single function?

Can some connected region in $\Bbb R^n$, possibly with some other nice conditions on the region, always be parametrized by a single function $\vec r(x_1, x_2, \dots, x_k)$ (even if it may be easier to ...
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1answer
27 views

Gradient of an implicitely defined function?

For some function $F(x,y,z) = 0$, is the gradient $\nabla F $ always equal to zero? If you take the partial derivatives of both sides, you get zero for all of them. My book says: Which implies that ...
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1answer
28 views

Lagrange Multipliers

The Question: Find the minimum distance between the origin and the surface $x^2y -z^2 +9 = 0$. I've been able to find the critical points when $x =0$ and when x is not equal to zero but lamda is ...
0
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1answer
26 views

Is this function of 2 variables differentiable?

$f(x,y) = \frac{\sin(x^4+y^4)}{x^2+y^2}$ when $(x,y) \neq (0,0)$ and $0$ when $(x,y) = (0,0)$ Is f differentiable?
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2answers
41 views

Chain rule for multiple variables?

What I've tried so far: $$F(x,y,z(x,y)) = 0$$ $$\implies \frac{\partial F}{\partial x} = 0$$ By the chain rule: $$\frac{\partial F}{\partial x} = \frac{\partial F}{\partial z}\frac{\partial ...
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2answers
43 views

Contradiction when differentiating?

Consider the function $F = x+y$. Let $x = t$ and $y= \cos t$. By directly differentiating, $$\frac{\partial F}{\partial x} = 1$$ and $$\frac{\partial F}{\partial y} = 1$$ Using the chain rule ...
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0answers
80 views

Question about a proof in Lang's undergraduate analysis

This is from page 580 of Lang's undergraduate analysis (2nd edition). I have difficulty in understanding the proof, hope that someone here can enlighten me. My questions are: i) On line 5 of the ...
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1answer
31 views

Geometric interpretation of derivative?

For some function $F(x,y) = 0$, $$\frac{dy}{dx} = \frac{-F_x}{F_y}$$ Can someone give me a geometric interpretatio of this? ($F_x$ and $F_y$ are the partial derivatives)
2
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2answers
64 views

How to check extrema if second derivative test fails

I have to find minima and maxima of $f(x,y)=x^4+6y^2-4xy^3-1$ I found three points that could be extrema - $(0,0)$, $(1,1)$ and $(-1,-1)$ I already checked $(1,1)$ and $(-1,-1)$ with second ...
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0answers
326 views

Finding the volume between a paraboloid and plane.

I have to find the volume between the plane $z=3-2y$ and the paraboloid $z=x^2+y^2$. I understand that the domain of this region is the disk centred at (0,-1) with radius 2 if I set the equations ...
2
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0answers
130 views

Pullback of Euclidean metric under spherical coordinates

I came across the following problem when I was reading a paper. Let $\Omega=B_2\subset\mathbb{R}^3$, which is the ball of radius 2 centered at origin. Map $F:B_2\backslash \{0\}\to B_2\backslash B_1$ ...
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2answers
76 views

How to show that the curve $ (x,y,z) = \langle \cos t, \sin t, c\sin t\rangle $ is an ellipse?

Show that the curve $$(x,y,z) = \langle \cos t, \sin t, c\sin t\rangle $$ is an ellipse in the plane it lies on. $$x^2 + y^2 = (\sin t)^2 + (\cos t)^2 = 1$$ $$x^2 + (z/c)^2 = (\sin t)^2 + (\cos ...
0
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2answers
41 views

Derivation for the integrating term in line integrals and volume integrals in spherical coordinates

Can anyone refer me to, or respond with, the derivation for the integrating term in line integrals $dl=dr\hat{r}+rd\theta\hat{\theta}+r\sin\theta\ d\phi\hat{\phi}$ and volume integrals ...