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).

learn more… | top users | synonyms

0
votes
0answers
10 views

Proving that average value of $u$ around a circle is the value of $u$ at the centre.

I would like to prove that: If $u(x,y)$ is harmonic in a domain containing a disk of radius $r$ with boundary $C_r$ $\implies$ the average value of $u$ around the circle is the value of $u$ at the ...
2
votes
1answer
33 views

Trying to integrate $\iint_D x^4\tan(x)+3y^2 \,dA$.

I'm trying to integrate $\iint_D x^4\tan(x)+3y^2\, dA$ in domain $D=\{(x, y) \in \Bbb R^2 \mid x^2+y^2\le4, y\ge0\}$. Domain is simple enough; half circle of radius 2 over $x$ axis. Converting to ...
0
votes
0answers
14 views

Partial derivatives after a change of variables

Say I have a function of $n$ variables $F(x_{1}, x_{2}, x_{3},...,x_{n})$, where $x_{1} = g_{1}(y_{1}, y_{2}, y_{3},...,y_{m})$, $x_{2} = g_{2}(y_{1}, y_{2}, y_{3},...,y_{m}),\dots, x_{n} = ...
0
votes
0answers
18 views

Maximizing the following function

I need to find values of $k_1$, $k_2$ and $k_3$ that maximize $C^{m_1}_{mm_1} \cdot C^{m_2}_{k_1-mm_1} \cdot C^{n_1}_{nn_1} \cdot C^{n_2}_{k_2-nn_1} \cdot C^{p_1}_{pp_1} \cdot C^{p_2}_{k_3-pp_1}$ ...
2
votes
0answers
49 views

Understanding a step in Yi Fang's Lectures on Minimal Surfaces

In Yi Fang's Lectures on Minimal Surfaces, page $94$, there's a step that I didn't understand, and that perhaps is wrong. I'll estabilish some notation first. We have that $X$ is a minimal surface, ...
1
vote
3answers
46 views

Prove that $f(x,y)$ is continuous in $(0,0)$

Prove that $f(x,y)$ is continuous in $(0,0)$, where \begin{equation} f(x,y) = \begin{cases} \frac{x^2y}{x^4+y^2}, & (x,y)\neq 0\\ 0, & (x,y) = (0,0) \end{cases} \end{equation} The solution I ...
0
votes
0answers
10 views

formula to establish correlation between multiple library functions

I am trying to predict the change in timeliness of holds delivery relative to number of owned Bestsellers, number of holds and the checkout window. (yes, it really is a library question). To do this ...
0
votes
0answers
22 views

How to prove the relation $\tan \theta = \hat{\vec{n}} \cdot \nabla h$

The relation for finding the contact angle is often given as $\tan \theta = - \hat{\vec{n}} \cdot \nabla h$ in papers such as in Sequential deposition of overlapping droplets to form a liquid line ...
1
vote
0answers
24 views

Find the mass flow rate, given a surface, density and velocity field

I have a confusion, I hope you can help me (I'd like that if you will respond, please read all my post). They ask me to find the mass flow rate passing through a surface, where the velocity field is ...
1
vote
0answers
9 views

Finite cover of balls for a bounded subset

Suppose X is a bounded subset of a k-dimensional manifold $M$ $\subset$ $R^n$ with $(k-1)$-dimensional volume $0$. I need to show that this implies for all $d>0$, all points of X can be contained ...
1
vote
1answer
47 views

Vector calculus identity for $\nabla\times(\vec{b}\cdot\nabla)\vec{b}$

I'm going through a paper on turbulence and in it the author uses the following $$ ...
1
vote
2answers
30 views

Convergence of a double integral

Is the integral $$\int_1^\infty\int_{e^{-x}}^1\frac{\sin y}{x^2y}dy dx$$ convergent or divergent?
0
votes
0answers
28 views

surface integral of the normal component of $curl(\mathbf{F})$

I have a square with side lengths $a$ in the $xy$-plane, and we build he square up vertically in the $z$-direction to construct a cube. What is the surface integral of the normal component of the ...
1
vote
1answer
24 views

Finding domain, range, and level curves of $f(x,y)=\arcsin(6y-5x)$

I will like it if someone helps me out and also checks my work for me for this equation: $f(x,y)=\arcsin(6y-5x)$ Domain $-1\leq6y-5x\leq1$ $y\geq\frac{5x}{6}-\frac{1}{6}$ and ...
2
votes
1answer
33 views

$f(x,y)=4x^3y^2$ Directional Derivative…

Let $f(x,y)=4x^3y^2$ How do I find the directional derivative of $f$ at $(2,1)$ in the direction of the vector $3i-4j$? What would be a unit vector in the direction in which $f$ decreases most ...
2
votes
3answers
63 views

Finding Extrema of $f(x,y)=x^4+y^4-4xy$

Let $f(x,y)=x^4+y^4-4xy$ How do I find all the relative extrema and saddle points of $f$ which lie within the open square ${(x,y) | -2<x<2,-2<y<2}$. And also if $f$ was in the closed ...
1
vote
0answers
15 views

$f(x,y)=4x^3y^2$ Dealing with Directional Derivatives and Vectors

Let $f(x,y)=4x^3y^2$ How do I find the directional derivative of $f$ at $(2,1)$ in the direction of the vector $3i-4j$? What would be a unit vector in the direction in which $f$ decreases most ...
0
votes
1answer
28 views

Proving that $f(x)=0\ \forall x\in B(0,r)$

Let $y=f(x_1,…,x_n)$ be differentiable on $B(0,r)$. Assume that $\dfrac{\partial}{\partial x_i}f(x)=0\ \forall x\in B(0,r)$ and $i\in\{1,…,n\}$. How to prove that $f(x)=0\ \forall x\in B(0,r)$? Do ...
-1
votes
0answers
23 views

Computing $f_{xy}$ and $f_{yx}$ [duplicate]

Let's consider the following function: $$f(x,y)=\begin{cases} xy\left(\dfrac{x^2-y^2}{x^2+y^2}\right) & \text{if }x^2+y^2\not=0 \\{}\\ 0 & \text{if }x=y=0 \end{cases}$$ I need to compute ...
0
votes
2answers
27 views

$f(x,y)=x^3+3xy^2-2y^3$. Find all unit vectors, if any, such that $f_u(0,1)=\frac{6}{5}$

I think that I understand what the question wants me to do: $f(x,y)=x^3+3xy^2-2y^3$. Find all unit vectors, if any, such that $f_u(0,1)=\frac{6}{5}$ I worked out the partial derivatives: ...
1
vote
2answers
20 views

Plot the level curve of sine function in multiple variables

I'm very confused about how I could go about this, as it seems that the question cannot be done using only the information given. The question is: plot the level curve for $f(x,y) = \sin(k^2x^2 + ...
1
vote
1answer
55 views

Proving that $f$ is differentiable at $0$

Let's consider the following function: $$f(x,y)=\begin{cases} (x^2+y^2)\sin\left(\dfrac{1}{x^2+y^2}\right) & \text{if }x^2+y^2\not=0 \\{}\\ 0 & \text{if }x=y=0 \end{cases}$$ I know that ...
1
vote
2answers
40 views

Find a unit vector that is parallel to $\nabla f(\cos\theta,\sin\theta)$

Suppose $f(x,y)$ is differentiable for all $(x,y),f(x,y)=17$ on the unit circle $x^2+y^2=1$, and $\nabla f$ is never zero on the unit circle. For any real number $\theta$, I have to find a unit vector ...
1
vote
0answers
18 views

Volume of the solid between these two parabaloids - can someone verify my answer?

I wish to find the volume of the solid bounded by the surfaces: $z = x^2 + y^2$ and $z = 4 - a^2x^2 - a^2y^2$. I set the two surfaces equal and it gave me a circle, so, I used cylindrical ...
0
votes
0answers
32 views

Find Directional Derivative, Unit Vector, and Rate of Change

Let $f(x,y)=4x^3y^2$ How do I find the directional derivative of $f$ at $(2,1)$ in the direction of the vector $3i-4j$? What would be a unit vector in the direction in which $f$ decreases most ...
3
votes
1answer
25 views

Lagrange multipliers method - absolute maximum and minimum

Using the Lagrange multipliers method I have to find the absolute maximum and minimum value of $f(x, y)=x^2+y^2-x-y+1$ in the unit disc. So, I have to find the extremas of $f(x, y)=x^2+y^2-x-y+1$ ...
2
votes
2answers
57 views

How to evaluate $\sum_{j=0}^\infty\;\sum_{\substack{k=0 \\ k \neq j}}^\infty \frac{1}{j^2-k^2}$

I was reading an introductory text on multiple integrals and I have encountered a problem asking me to explain why $$ \sum_{j=0}^\infty\;\sum_{\substack{k=0 \\ k \neq j}}^\infty ...
2
votes
1answer
18 views

Determine the volume of a solid given specific bounds

Determine the volume of the solid enclosed by the paraboloid $z = x^2 + y^2$ and the plane with equation $4x − 2y + z = 0$. Could someone explain to me whether I use double integral polar coordinates ...
0
votes
1answer
16 views

Parameterise the path C of a square

I have a question, I am required to parameterise the square with side lengths $a$, going in a counterclockwise direction. I have determined then that the points are $$(0,a), (a,0), (0,0), (a,a)$$ ...
6
votes
0answers
62 views

Very difficult surface integral

Compute the surface integral: $$\int_S({x\over \sqrt{x^2+y^2+z^2}}, {y\over \sqrt{ x^2+y^2+z^2}}, {z\over \sqrt{x^2+y^2+z^2}}), \cdot \vec n \ dS$$ where $S: x^3+y^3+z^3=a^3$ The first ...
1
vote
2answers
50 views

Are partial derivatives a special case of the total derivative or just something else entirely?

I can do basic multivariable calculations using partial and total derivatives. I also know for partial derivatives the existence of all partial derivatives at a point doesn't imply continuity. Are ...
1
vote
0answers
5 views

Conservative field $F$ on not simple connected set

Give an example of a field $F:D\subseteq \mathbb R^2 \to \mathbb R^2$ such that $D$ is a doubly connected set (that is $D$ has on "hole") but $F$ is conservative And if $D$ is a triply connected ...
1
vote
0answers
34 views

Prove or Disprove that if $f_x(x_0,y_0)$ and $f_y(x_0,y_0)$ both exist, then f is continuous at $(x_0,y_0)$ [duplicate]

I have to either prove or disprove the fact that if $f_x(x_0,y_0)$ and $f_y(x_0,y_0)$ both exist, then f is continuous at $(x_0,y_0)$. What I thought: I thought that the best way to approach this is ...
0
votes
1answer
31 views

How does one read $\Delta \mathbf{E} = (\mathbf{d}\cdot \nabla ) \mathbf{E}$

I'm reading a book on Electrodynamics and came across this formula: $$ \Delta \mathbf{E} = (\mathbf{d}\cdot \nabla ) \mathbf{E} $$ where $\Delta \mathbf{E}$ represents the difference (delta) in an ...
2
votes
1answer
18 views

Prove that $g_x(x,y,z)+g_y(x,y,z)+g_z(x,y,z)=0$

I am having some trouble proving the following: Prove that if $f$ is a differentiable function of $3$ variables and $g(x,y,z)=f(x-y,y-z,z-x)$, then $g_x(x,y,z)+g_y(x,y,z)+g_z(x,y,z)=0$ I tried ...
2
votes
1answer
40 views

A counterexample for a smoth version of Tietze extension theorem

Is there any function $f:F\subset \mathbb{R}^2\rightarrow \mathbb{R}$ with $F$ closed such that $f|F$ is differentiable in every accumulation point but there is no differentiable extension to the ...
2
votes
0answers
27 views

Finding points on surface with specified tangent planes

Suppose that $ f(x,y,z) = \frac{3}{2-x} + \frac{1}{y-z} $ and let $S$ be the surface given by the equation $f(x,y,z) = 1$ Are there any points on $S$ where the tangent plane to $S$ is parallel to ...
2
votes
2answers
35 views

Determining what set of points a curve can be expressed as a singlevariable-function

The curve $$x^2y^3-3xy^2-9y+9=0$$ is given. I want to determine what points on the curve, for a neighbourhood to said points, $y$ can safely be expressed as a function of $x$. I guess what this means ...
0
votes
0answers
22 views

Gradient of a function at the boundary of a constant region

Seemingly an easy thing to do, I had difficulty to find an answer for the following: Let's assume we have a function $f(x)$ which is defined as $f:\mathbb{R^n} \to \mathbb{R}$. The function has ...
7
votes
1answer
87 views

Does the inverse function theorem hold over $\mathbb{Q}$?

Let $f:\mathbb{Q}^n \longrightarrow \mathbb{Q}^n$. We can define what it means for such $f$ to be differentiable: (The differential will be a linear transformation $\mathbb{Q}^n \longrightarrow ...
2
votes
0answers
30 views

Why is the negative of the gradient the direction of greatest descent?

I imagine it as if one is going up a physical hill. It doesn't seem like there's a guarantee that going in the opposite direction of greatest increase in height will necessarily be the direction of ...
0
votes
3answers
40 views

Stuck on a Limit Question in Multivariable Calculus [duplicate]

I have just started learning about limits in my multivariable class and I came to a problem: Let $$h(x,y)=\frac{x^5y}{2x^{10}+y^2}.$$ How would I prove that $$ \lim_{(x,y) \to (0,0)} h(x,y) \text{ ...
2
votes
1answer
39 views

How do I differentiate a Kronecker product with respect to a vector?

I am trying to differentiate $[\mathbf{I} \otimes \mathbf{t}^*\mathbf{t}^T]$ with respect to $\mathbf{t}$. I did the following $\mathbf{I} \otimes \mathbf{t}^*\mathbf{t}^T = (\mathbf{I} \otimes ...
2
votes
1answer
47 views

Calculating the distance function and projection

Let $\Omega=\{x\in\mathbb R^n\mid\langle a,x\rangle=b\}$. We define the distance function and projection as follows $$d(x;\Omega)=\inf\{||x-\omega||\mid\omega\in\Omega\}$$ ...
0
votes
1answer
29 views

Prooving Kepler's Second Law through vectors.

I am taking a multivariable calculus lecture online provided by MIT OpenCourseWare. ...
0
votes
2answers
36 views

Total Differential / Ito dynamics

I found this process in a scientific paper: $M_t = \int_{0}^t e^{-(t-u)} \frac{dS_u}{S_u}$ where $dS_t = S_t (\phi M_t + (1-\phi)\mu_t) dt + \sigma S_t dW_t$ and I want to compute the ...
0
votes
1answer
23 views

An application of Greens's theorem

Apply Green's theorem to prove that, if $V$ and $V'$ be solutions of Laplace's equation such that $V=V'$ at all points of the closed surface $S$, then $V=V'$ throughout the interior of $S$. ...
0
votes
0answers
25 views

When is system of linear equations smooth

I am wondering when is a system of linear equations smooth? More specifically, for Ax=B, what property of A guarantees smoothness of systems of linear equations? If it is known that A is always ...
0
votes
1answer
25 views

Find Formula for Curvature and Max and Min Values

If $C$ is a smooth curve in 3-space parametrized by arc length, then then curvature $k(t)$ is defined as $$k(t)=\frac{||r'(t)\times r''(t)||}{||r'(t)||^3}$$ Let $C$ be the curve parametrized by ...
1
vote
3answers
39 views

Definition of Limit in Multivariable Calculus

Let $f(x,y)$ be a function defined in some disk that is centered at $(x_0,y_0)$. Suppose that $L$ is some real number. Then in this case, what does it mean that: $$\lim_{(x,y)\to (x_0,y_0)} f(x,y)=L$$ ...