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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13
votes
4answers
312 views

How prove this integral limit $=f(\frac{1}{2})$

Let $f$ be a continuous function on the unit interval $[0,1]$. Show that $$\lim_{n\to\infty}\int_{0}^{1}\cdots\int_0^1\int_{0}^{1}f\left(\dfrac{x_{1}+x_{2}+\cdots+x_{n}}{n}\right)dx_{1}dx_{2}\cdots ...
3
votes
1answer
43 views

Continuity in $\mathbb R^n$.

we just got started with this topic today, and I am confused. Let $f:\Bbb R^2 \to \Bbb R $ with $$f(x,y) =\begin{cases} y\sin(x)/x &\text{if } x \ne 0\\ 0 &\text{else} \end{cases}$$ Now, ...
1
vote
1answer
28 views

What are $a$ and $b$ when the zeropoints of $f(z)=(a+bi)z+2-i=0$ is at $1-i$?

$f(z)=(a+bi)z+2-i$. What are the values of a and b when $1-i$ is the zeropoint of f? $f(z)=(a+bi)z+2-i=0$ $(a+bi)(1-i)+2-i=0$ $a+bi-ai-bi^2+2-i = 0$ $(a+b+2)+(-a+b-1)i=0$ I don't know what the ...
0
votes
1answer
39 views

Find the absolute maximum and minimum values of f(x,y)

Find the absolute maximum and minimum values of f on the set D: $f(x,y)=(x-y)(1-x^2-y^2)$ $D=\left\{(x,y) \mid x^2+y^2\le1\right \}$ Can someone help me resolving the system of partial ...
0
votes
1answer
135 views

Lipschitz continuity of $f(x,y)=4x^2+xy-\frac{1}{y-1}$ on an open set $U \subset \mathbb{R} \times (\mathbb{R} \setminus \lbrace 1 \rbrace)$

Problem: Find an open set $U \subset \mathbb{R} \times (\mathbb{R} \setminus \lbrace 1 \rbrace )$ which includes the points $(0, 1/2$) and $(0,3/2)$ such that the function ...
0
votes
0answers
49 views

Hyperbolic distance

Find the hyperbolic distance between $(0; 0; 0)$ and $(0; 0; \frac12)$ in the Poincare model. Recall that the Poincare model deems $d(P_1; P_2)=\int\frac{2}{1-r^2}ds$. What about the distance between ...
2
votes
1answer
118 views

Tangent planes perpendicular at each point of intersection

Find the set of all points $(a,b,c)$ in 3-space for which the two spheres $(x-a)^2+(y-b)^2+(z-c)^2=1$ and $x^2+y^2+z^2=1$ intersect orthogonally.( Their tangent planes should be perpendicular at each ...
1
vote
2answers
83 views

Aside from this two practical technique to compute any integral, what else? [closed]

Aside from this two practical technique to compute any integral, what else could called a fundamental method but not approximate method like Riemann Sum? These two method I've been referring to are ...
1
vote
0answers
38 views

Equation of tangent plane to a surface at certain point

So I am solving for an equation of a tangent plane to surface $z=x + \ln(2x+y)$ at the point $(-1,3,-1)$. I know I need to take partial derivative of that equation and plug in the point of the ...
2
votes
2answers
38 views

Proving a vector identity

Let $\vec{a} , \vec{b} , \vec{c} $ three nonzero, non parallel vectors in $\mathbb{R}^3 $ for which $ (\vec{a} \times \vec{b} ) \times \vec{c} =\vec{0} $ . Prove that $\vec{a}\cdot \vec{c} = ...
2
votes
1answer
142 views

Double integral region

I have made an attempt on a problem from an old exam. I'm not sure if my method is correct or not, as it differs from the teacher's solution and I'm unsure of the theory. Does my solution lack any ...
1
vote
1answer
402 views

Calculating the center of mass in spherical coordinates

So normally, to calculate the center of mass you would use a triple integral. In my particular problem, I need to calculate the center of mass of an eight of a sphere where it's density is ...
1
vote
1answer
22 views

$\iint_D \frac{1}{1+x^2+y^2} dx dy$

Problem: $$\iint_D \frac{1}{1+x^2+y^2} dx dy,$$ where $D=\left\{(x,y):0 \le x \le y\right\}$. I got everything right, except the region. The book said $\pi / 4 \le \theta \le \pi /2$ and I wanted the ...
2
votes
1answer
84 views

Gradient Vector Question?

The temperature in some three-dimensional body is modeled by the equation $$f(x,y,z)=49-x^2-y^2-z^2$$ Find the largest rate at which the temperature is increasing when T=0. I believe this is a ...
1
vote
1answer
59 views

Tangent Planes and Surfaces (Calc 3)

I am wondering if I am on the right track for the following question: Find a for the plane $x+y+z=-1$ so that it is a tangent plane to the surface $z=x^2+ay^2$ I figured since you are given a ...
1
vote
1answer
34 views

Prove that there exists only one function f such that…

Prove that there exists only one function $$\big[f\in C\left ( \left [ 0,1 \right ],\mathbb{R} \right )s.t. f(x)=\frac{2}{5}\int_{0}^{1}(x^{2}+t^{5})f(t)dt+sin(x)\big] $$
0
votes
2answers
57 views

Trouble seeing how Lagrange Multipliers are True

So if a function $f:\mathbb{R}^n\rightarrow\mathbb{R}$ constrained to the surface $g(x)=c$ for $x\in\mathbb{R}^n$ has a local maximum at $P$, then I'm having trouble seeing how this implies that the ...
2
votes
1answer
48 views

Surface Area Line integral problem

I'm trying to figure out how to solve a surface area with surface and line integrals (showing both methods). The area I'm trying to compute is the area of the shape $$x^2+y^2=9$$ bounded by $z=0$ and ...
0
votes
0answers
237 views

Finding surface integral of the paraboloid and disk

Let S be the surface consisting of the paraboloid $y=x^2 + z^2$ with $0 \leq y \leq 1$, and the disk $x^2 + y^2 \leq 1$. Let $S$ have an outward orientation. Compute the double integral of $\langle ...
0
votes
1answer
24 views

How to find the line integral from $(0,0)$ to $(1/8,0)$

I have to find the line integral from $(0,0)$ to $(1/8,0)$. I just want to know if what I did is correct. $x=x$ and $y=0$ where $x$ is less than or equal to $0$ and greater than or equal to $1/8$. ...
3
votes
3answers
154 views

Lagrange multipliers from hell

I was asked to solve this question, decided to try and solve it with lagrange multipliers as I see no other way: "Find the closest and furthest points on the circle made from the intersection of the ...
1
vote
0answers
81 views

Parametrizing a “surface” that is actually a curve, then integrating to find the area?

Find a parametrization of the surface $x^2-y^2=1$, where $x>0$, $-1 \leq y \leq1$ and $0 \leq z \leq 1$. Use your answer to express the area of the surface as an integral. I'm confused because ...
0
votes
2answers
51 views

Integrating $g: ℝ^2\to ℝ$ - Order of Integration

The problem: My work: I found the two integrals to be equal to each other, which is clearly not the desired result. Any suggestions/pointers? Thanks!
2
votes
0answers
66 views

Green's Theorem proof using Stoke's theorem

I'm slightly confused with this proof, from Stoke's Theorem we have: $$\int_C \underline{F} \cdot \ d \underline{r} = \int \int_S (\nabla \times \underline{F}) \cdot \underline{n} \ dS$$ so going ...
0
votes
2answers
105 views

Surface Area of Two Cylinders Calculus 3

Find the surface area of two cylinders $$y^2 + z^2 = 1$$ and $$x^2 + y^2 = 1$$ I have so far set the two equations to equal $$x= \pm z$$ and $$y= \sqrt{(1-z^2)}$$ I am a little confused on how to set ...
0
votes
1answer
56 views

particular solutions is zero

We have some issues with particular solution. Cannot solve A and B on the last line because it becomes zero all together. So it becomes $2sin(2x) = 0$ What are we doing wrong? Thanks for your time. ...
3
votes
0answers
175 views

Is there a generalization of integration by parts?

In the original integration by part formula there are two functions $u(x)$ and $v(x)$. What if the integral involves another function $w(x)$ as well? Second of all, I know that there is a several ...
0
votes
0answers
45 views

Chain rule for several variables

I am studying this example: I follow the first two statements, but I cannot make the connection between the dot product and the derivative. Can somebody please explain how the third equation ...
2
votes
2answers
83 views

Total derivative

What is the significance and meaning of the total derivative? Why is it introduced in the definition of differentiability of scalar and vector fields?
0
votes
0answers
67 views

predictor-corrector method and stability

A predictor-corrector method for the approximate solution of $y'=f(t,y)$ uses \begin{equation} y_{n+1}-y_{n}=hf_{n} \tag P \end{equation} as predictor and \begin{equation} ...
1
vote
0answers
24 views

How to set up a surface integral on a cylinder cut by planes?

Find the surface area of the piece of the cylinder $x^2 + y^2 = 4$ cut off by the planes $z = 0$ and $y = z$ with $y \ge 0$ using surface integrals. Can someone help me set up this surface integral ...
3
votes
1answer
39 views

Multivarable limit proof

I came across with this statement and I can't neither prove it right nor find a counterexample. The statement is: Consider two functions $F(x,y)$ and $G(x,y)$ continuous and differentiable around a ...
0
votes
1answer
51 views

In this problem, what am I taking the integral of?

I'm a little confused by this problem because I have no idea what I should be taking the integral of. I was following a book example when I realised that the book explicitly tells you what to take the ...
0
votes
2answers
125 views

Line integral over a curve in the II quadrant

I am lost here: $C = x^2 + y^2 = 4$ from $(0,2)$ to $(-2, 0)$. Calculate $ \ \int_c y^2 ds \ \ $ and give reasons the sign is correct. It's obviously the circular arc going counterclockwise from ...
0
votes
2answers
103 views

Calculating partial derivative for a function defined by an integral

I have no idea how to solve the following problem. Please suggest some suitable solutions. Define $$f(x,y)=\int_0^{\sqrt{xy}} e^{-t^2} \,dt,$$ for $x>0, y>0$. Compute $\dfrac{\partial ...
20
votes
5answers
3k views

Limit is found using polar coordinates but it is not supposed to exist.

Consider the following 2-variable function: $$f(x,y) = \frac{x^2y}{x^4+y^2}$$ I would like to find the limit of this function as $(x,y) \rightarrow (0,0)$. I used polar coordinates instead of ...
2
votes
1answer
51 views

A question on gradients: $f$ must assume equal value at two points

If $\nabla f(x,y,z)$ is always parallel to $x i+y j+z k$, show that $f$ must assume equal values at the points $(0,0,a)$ and $(0,0,-a)$.
0
votes
1answer
45 views

k+1 Differential form

Consider the k-form given by, $ w = \sum_{i_{1}<i_{2}<...<i_{k}} w_{i_{1}i_{2}...i_{k}} dx^{i_{1}}\wedge dx^{i_{2}}\wedge...\wedge dx^{i_{k}}$ Define $k+1$ form $dw$ , the differential of ...
0
votes
1answer
291 views

How to find an equation of the plane, given its normal vector and a point on the plane? [duplicate]

I have a question regarding vectors: Find the equation of the plane perpendicular to the vector $\vec{n}\space=(2,3,6)$ and which goes through the point $ A(1,5,3)$. (A cartesian and parametric ...
1
vote
1answer
47 views

Implicit Euler Scheme and stability

Find the fixed points of the implicit Euler scheme \begin{equation} y_{n+1}-y_{n}= hf(t_{n+1},y_{n+1}) \end{equation} when applied to the differential equation $y'=y(1-y)$ and investigate their ...
2
votes
1answer
95 views

Verify Stokes's Formula for…

Verify Stokes's Formula for $\textbf{F}(x,y,z)=(3y,-xz,yz^2)$, where $S$ is the surface of the paraboloid $2z=x^2+y^2$ bounded by the plane $z=2$. So I need to compute the integral using the formula ...
2
votes
1answer
65 views

Proving a set of $2\times 3$ matrices is a manifold?

The way I have always been told to check if something is a manifold (I haven't had a whole lot of experience with them), is to check if the derivative of the function representing the loci of the ...
3
votes
2answers
69 views

Computing a Lie Bracket: General Questions

I'm asked to compute the following Lie Bracket: $\left [ -y \dfrac{\partial}{\partial x} + x\dfrac{\partial}{\partial y} , \dfrac{\partial}{\partial x} \right] $ on $\mathbb{R}^2$. Just writing it ...
5
votes
1answer
216 views

Directional, differential and lie derivatives on manifolds intuition?

Trying to translate elementary multivariable calculus into the language of manifolds: Is the directional derivative on a manifold just a way of finding the rate of change of a vector in a single ...
2
votes
2answers
31 views

Application of chain rule

The equations $u=f(x,y),x=X(t),y=Y(t)$ define $u$ as a function of $t$, say $u=F(t)$. Compute $F'(t)$ in terms of $t$ if, $$f(x,y)=\log [(1+e^{x^2})/(1+e^{y^2})] , X(t)=e , Y(t)^t=e^{-t}.$$ From the ...
1
vote
0answers
39 views

Understanding optimization on non-compact region

Say we have $f(x,y) = x^2 e^{-x^2 - y^2}$ and we want to optimize it over $\mathbb{R}^2$. The minimum value is $0$ since $f(x,y) \geqslant 0$; the question is whether a maximum value exists or not. ...
0
votes
2answers
27 views

Polar coordinates and Jacobian of $\frac12 r $

To solve a double integral problem, I just did the sub $$x = \frac12 r \cos( \theta ) , \quad y = r \sin( \theta )$$ and the Jacobian is $\frac12 r $ but I realise – I'm not sure how to write that ...
6
votes
5answers
474 views

Why aren't these partial derivatives interchangeable?

I've ran across something that confuses me regarding multivariable functions and partial derivatives. I'll use an example to illustrate: We let $$x = f(y,t) = yt^2,$$ and define the operators ...
0
votes
2answers
85 views

Prove that $f'_{xy}=f'_{yx}$ [duplicate]

Here is a basic, and probably a bad, question. A fundamental rule of derivatives. Why is $f'_{xy}=f'_{yx}$ true?
0
votes
0answers
28 views

Nonexistence of a scalar field

Prove that there is no scalar field $f$ such that $f'(a;y)>0$ for a fixed vector $a$ and every non-zero vector $y$.