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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19 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?
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0answers
15 views

surface integral of the normal component of curl F

If I was given a square with side lengths $a$ in the $x-y$ plane. If the square were built up vertically in the z-direction to construct a cube, what would be the surface integral of normal component ...
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1answer
21 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
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1answer
30 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
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3answers
48 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 ...
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0answers
13 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 ...
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1answer
26 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 ...
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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 ...
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2answers
25 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: ...
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1answer
8 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 + ...
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1answer
52 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
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2answers
31 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 ...
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0answers
15 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 ...
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0answers
30 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
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1answer
22 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
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2answers
52 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
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1answer
15 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 ...
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1answer
15 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 $$(0,a), (a,0), (0,0), (a,a)$$ ...
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0answers
61 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 ...
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2answers
44 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 ...
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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 ...
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0answers
33 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 ...
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1answer
28 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
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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
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1answer
39 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
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0answers
26 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
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2answers
34 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 ...
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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 ...
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1answer
81 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
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0answers
29 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 ...
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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
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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
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1answer
45 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 $$d(x;\Omega)=\inf\{||x-\omega||\mid\omega\in\Omega\}$$ ...
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1answer
26 views

Prooving Kepler's Second Law through vectors.

I am taking a multivariable calculus lecture online provided by MIT OpenCourseWare. ...
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2answers
35 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 ...
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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 $S$. ...
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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 ...
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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 ...
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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: $$\lim_{(x,y)\to (x_0,y_0)} f(x,y)=L$$ ...
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1answer
44 views

Find the volume of the solid in $\Bbb R^3$

I need to find the volume of the solid in $\Bbb R^3$. It is bounded by the following: $y=x^2$, $x=y^2$, $z=x+y+21$ and $z=0$. I known that the volume is expressed as follows: $$\iiint 1 \, dV$$ I ...
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2answers
31 views

Tangent Plane to the Surface $x^3y+2y^3z+3xz^3=16$ at the Point $(0,2,1)$

Find an equation for the tangent plane to the surface $x^3y+2y^3z+3xz^3=16$ at the point $(0,2,1)$. I have a feeling that this includes partial derivatives in a way.
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1answer
13 views

Compute flux of vector field F through hemisphere

I need help solving this question from my textbook. Compute the flux of the vector field: $$\vec F = 4xz\vec i + 2 y\vec k$$ through the surface $S$, which is the hemisphere: $x^2 + y^2 + z^2 = 9 , ...
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2answers
60 views

Prove that the $\lim_{(x,y) \to (0,0)}h(x,y)$ Does not Exist using Polar Coordinates

Let $h(x,y)=\frac{x^5y}{2x^{10}+y^2}$. How would I prove that the $\lim_{(x,y) \to (0,0)}h(x,y)$ Does Not Exist? I think that we might use polar coordinates, but I am not sure.
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2answers
41 views

Prove that the $\lim_{(x,y) \to (0,0)}g(x,y)=0$

Let $g(x,y)=\frac{\sin^2(x-y)}{|x|+|y|}$. How would I prove that the $\lim_{(x,y) \to (0,0)}g(x,y)=0$? I was looking through my textbook and found that for all real numbers $s$ and $t$, ...
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1answer
25 views

Show that $T(t)$ and $N(t)$ are Orthogonal

If $r(t)$ is the smooth parametrization of a curve $C$ in 3-space, then the unit tangent and unit normal vectors are denoted as $T$ and $N$, and are given by: ...
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1answer
13 views

Representation of derivative by alinear map

Let $\phi:\mathbb R^2 \to \mathbb C$ be the map $\phi(x,y)=z$ where $z=x+iy$.Let $f:\mathbb C\to \mathbb C$ be the function that is $f(z)=z^2$and $F=\phi^{-1}f\phi $.. Represent the derivative of ...
3
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1answer
17 views

Finding the parameterization of a curve for a line integral problem

I have to calculate the work of a particle that travel along a curve, given the following vector field: $F(x, y, z) = (2z-1, 0, 2y)$ and where the curve is the intersection between: $s1: z = x^2 + ...
0
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3answers
48 views

Tangent to surface given a point [on hold]

so is the way I approach this : gradF(x,y,z)?
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0answers
19 views

Computing gradient using implicit functions theorem

I am trying to compute analytically the gradient of a function to specify it in a Python program and find the minimum more rapidly. I did the computation many many times and I do not manage to find ...
0
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2answers
24 views

evaluating the double integral

I tried to calculate $\int _0^9 dx\:\int _{-\sqrt{x}}^{\sqrt{x}}\:y^2dy$ which yielded $c$ as in this integral has no particular value...when I plot the graphs for it's D however, a certain area does ...