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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3
votes
1answer
21 views

introduction to potential theory in $\mathbb{R}^3$ [on hold]

A differentiable function $g: \mathbb{R}^3 \to \mathbb{R}$ is said to be harmonic in a subset $B \subset \mathbb{R}^3$ if $\Delta^2 g = 0$ for all $p \in B$. Let $M \subset \mathbb{R}^3$ be a bounded ...
0
votes
1answer
45 views

Finding points that satisfy $f(a) = \sup f(x)$

Choose positive real numbers $\alpha_1,\ldots,\alpha_n$, $n$ such that $\sum_{i=1}^n \alpha_i = 1$ and let $$f: [0,\infty)^n \to \mathbb R$$ $$x=(x_1, \ldots, x_n) \mapsto x_1^{\alpha_1} \cdots ...
4
votes
1answer
36 views

Gradient in polar coordinates

The gradient in the normal direction on a given contour of a function, expressed in polar coordinates as $u = u(r, \theta)$, can be calculated as: $$\frac{\partial u}{\partial n} = \frac{\partial ...
1
vote
2answers
29 views

Taking derivatives of the implied function - from the implicit function theorem,

I showed that the relation $$f(x,y)=e^x - e^y + xy = 0$$ defines near (0,0) an implicit function y=$\phi (x)$, since the $1x1$ block, $\frac{df}{dy}$, evaluated at (0,0) gives -1, which is non-zero - ...
0
votes
0answers
15 views

Additive & Multiplicative Link

Is there an isolative property about the following multivariable equation: $$f_1(x)g_1(y)=f_2(x)+g_2(y)$$ That is, is this equation rearrangable such that it may be put in the form ...
3
votes
1answer
23 views

Find the area of a subset of $\mathbb{R}^3$ given by an implicit relation.

Let x, y, z be real numbers and let $A = \begin{bmatrix} 1&x&x^{2} \\ 1&y&y^{2} \\ 1&z&z^{2} \end{bmatrix} $ Let S be the subset of $\mathbf{R}^{3}$ given by $S = \{ ...
1
vote
2answers
115 views

Equality of mixed partial derivatives

Is the following statement $$\frac{\partial^2 f}{\partial x \, \partial y}=\frac{\partial^2 f}{\partial y \, \partial x}$$ always true? If not what are the conditions for this to be true?
4
votes
2answers
30 views

Integration by Change of Variable

By using change of variable, $$x+y=(\surd2)u \text { and } y-x=(\surd2)v$$ Evaluate $$I=\iint(y-x)^2e^{-(x+y)^2}dv\,du$$ with $R$ bounded by $x=0,y=0,x+y=1$ After changing of variable, I get ...
2
votes
2answers
23 views

Help me to understand Line integral problem solution

I as thinking as to why theta has been taken from 0 to 2pi .it is not obvious from picture . Also Can it be done using stokes .Thanks
4
votes
0answers
45 views

Seperating single integral into an double integral.

Please refer to : How to prove that $\int_{0}^{\infty}\sin{x}\arctan{\frac{1}{x}}\,\mathrm dx=\frac{\pi }{2} \big(\frac{e-1}e\big)$ The answer by @Venus. What is the procedure in converting that ...
0
votes
2answers
28 views

derivative of domain of integration

Suppose $f(x,y)$ is a function $\mathbb{R}^2\times \mathbb{R}^2 \to\mathbb{R}$ and $\Omega(x)$ is a family of compact regions of the plane whose boundary curve $\gamma(s,x)$ varies smoothly in $x$. I ...
2
votes
2answers
26 views

Does being a local minimum imply a positive definite hessian?

If $p\in R^{m}$ is a local minimum of $F:R^{m}\rightarrow R$, then can we conclude that $\dfrac{\partial ^2F}{\partial x \partial x'}[p]$ is positive definite?
2
votes
0answers
63 views

How show the map $f:\mathbb R^2\rightarrow\mathbb R$, defined as $f(x,y)=x+y$ is continuous for all $(x,y)\in\mathbb R^2$?

Question: I want to show the map $f:\mathbb R^2\rightarrow\mathbb R$, defined as $f(x,y)=x+y$ is continuous for all $(x,y)\in\mathbb R^2$. Issue: I know how to prove this via the epsilon-delta way. I ...
3
votes
2answers
26 views

Optimization - find the dimensions of a box as functions of volume - minimal surface area

Had a basic calculus course exam today. This was one of the problems: We have a rectangular box of a given volume V. Present the width, height, and length of the box as functions of V so that the box ...
0
votes
2answers
35 views

How do I find this distance?

Find the minimum and maximum distances between the ellipse $x^2+xy+2y^2 = 1$ and the origin. This is what I've attempted so far: Maximize $x^2+y^2+z^2$ with respect to $x^2+xy+2y^2 = 1$. Using ...
2
votes
1answer
29 views

Using Lagrange multipliers to find the extrema of $f(x,y) = e^{2xy}$ subject to $x^2+y^2 = 16$

Find the maximum and minimum values of $f = e^{2xy}$ with respect to $x^2+y^2 = 16$. Using Lagrange multipliers, $\nabla f = \lambda\nabla g$. Therefore, the constraints are the following: ...
-1
votes
1answer
22 views

Tangent Vector at the point $(1,2,11)$ whose projection onto the $xy$-plane is parallel to vector $1/\sqrt{10}i+3/\sqrt{10}j$.

$$f(x,y) = x^3y^2 + 3x + 2y$$ The gradient of $f$ at the point $(1,2)$ is $15{\bf i} + 6{\bf j}$. ${\bf u} = \dfrac{1}{\sqrt{10}}{\bf i} + \dfrac{3}{\sqrt{10}}{\bf j}$ The Directional Derivative ...
1
vote
1answer
25 views

Construct a procedure which determines the location of the shadow of a rectangluar box.

I drew a 3d rectangular box on a coordinate plan consisting of x, y, and z. A procedure is to be created that will determine the location of the shadow of the box on one of the coordinate planes. I ...
1
vote
1answer
53 views

Calculate surface area of a F using the surface integral

Task Given: $$F := \{(x,y,z) \in \mathbb{R}^3 \mid (x,y) \in W,z=f(x,y)\}$$ Calculate the surface area using the surface integral: $i) \; f(x,y) := x+y \;\; and \;\; W := [12,31] \times ...
0
votes
1answer
28 views

Calculate surface area of a sphere using the surface integral

Given a sphere with: $$F := \{(x,y,z) \in \mathbb{R}^3 \mid x^2+y^2+z^2 = 1, x\le0\}$$ $$ \Rightarrow r = 1, \varphi = [\frac{\pi}{2}, \frac{3\pi}{2}], \theta = [0, \pi] $$ My Task is to calculate ...
0
votes
0answers
29 views

How to calculate the volume of a tetrahedron?

Suppose that $$ I=\iiint_{V}f(x,y,z)dxdydz $$ where $f(x,y,z)$ is a continuous function, $V$ is a tetrahedron whose vertices are $P(2,2,0), A(-2,0,0), B(0,0,2)$ and $C(1,1,3)$. I want to ask you how ...
4
votes
1answer
39 views

Can the Heat Equation be Averaged Over a Region?

I am doing a project for my partial differential equations class in which I am motivating the definition of a weak solution. To get started, I assumed that $T$ was a solution to $\nabla^2 T = \partial ...
2
votes
1answer
26 views

Calculate surface integral

I need some help with the following: Given $$f(x,y,z)=\left( \frac{-x}{(x^2+y^2+z^2)^{\frac{3}{2}}}, \frac{-y}{(x^2+y^2+z^2)^{\frac32}}, \frac{-z}{(x^2+y^2+z^2)^{\frac32}} \right),$$ calculate the ...
1
vote
1answer
21 views

How can you find the distance between two skew planes?

I understand that there is a unique line perpendicular to both planes and the length of that line is the distance between the planes but how would I go about finding the what the equation of that line ...
0
votes
0answers
37 views

To find the Maximum and minimum value of f over square

Given function $f = (x+y)^2 - (x+y) +1$ .I have to find maximum and value of $f$ over square with unit side in first octant in xy-plane. I calculated $f_x $ and $f_y $ both came out to be ...
0
votes
0answers
22 views

Geometric position of gradient on the surface in $\mathbb{R}^3$ and orthogonality to tangent of level curve

Given a function $f(x, y)\in C^1(\mathbb{R}^2)$ and its gradient $\nabla f(x, y) =(\frac{\partial f(x, y)}{\partial x}, \frac{\partial f(x, y)}{\partial y})$ which forms a vector field where each ...
0
votes
1answer
30 views

Partial derivative is bounded

Let $f(t,z)$ be a bounded (say by a constant $M$) continuous function on $\mathbb{R}_t \times \mathcal{U}$ where $\mathcal{U}$ is an open neighborhood of $0 \in \mathbb{C}_z$. Moreover, for each fixed ...
4
votes
1answer
33 views

Independence of function and its derivative in calculus of variations

It's common to see in calculus of variation that the integrand $f$ of functional $F[y]=\int f(y,y',x)dx$ is a function of $y,y'$ and $x$. Why do we regard the derivative $y'$ as an independent ...
3
votes
1answer
20 views

Chain rule notation for composite functions

Suppose I have a function $ f(x, y, g(x, y)) $ How would I express $ \frac{\partial f}{\partial x} $? Using the chain rule, you'd naturally come up with $ \frac{\partial f}{\partial x} + ...
1
vote
0answers
25 views

Finding the normal vector of a surface (Flux of a vector field n*dS expression)

This problem is practice for a final exam. Let S be the closed surface whose bottom face B is the unit disc in the $xy$-plane and whose upper surface U is the paraboloid $ z = 1 − x^2 − y^2 , z \geq ...
0
votes
2answers
37 views

Finding the mass of a cone using triple integral

I have a density $\rho(x,y,z) = 3-z$ and have converted my given information to form a triple integral equation for finding the volume of my cone in cylindrical coordinates and have found the volume ...
0
votes
1answer
36 views

Volume bounded by two solids

Can somebody help me get started in the right direction for this question involving volume? The question is "Find the volume of the solid region inside the hemisphere $x^2 + y^2 + z^2 =6, z<0$ but ...
0
votes
1answer
26 views

Minimize squared distance to origin from a paraboloid

I have to use Lagrange multilpiers to find the minimum distance from the paraboloid with equation $z = \left({x-1/}{\sqrt{2}}\right)^2 + \left({y-1/}{\sqrt{2}}\right)^2$ to the origin, and from this ...
1
vote
1answer
44 views

Evaluate $\int_0^1\int_x^1 e^{x/y} dy\,dx$

I need some help to solve the following: $$\int_0^1\int_x^1 e^{x/y} dy\,dx$$ I guess it is related with change of variable, but I can't figure out which one. Thanks in advance. Regards.
12
votes
3answers
244 views

parallelizable manifolds

I know a differentiable manifold $M$ of dimension $n$ is parallelizable if there exist (smooth of course) vector fields $\{X_i\}_{j=1}^n$ which are linearly independent in $T_pM$ at each point $p \in ...
1
vote
0answers
29 views

Second derivative of the position vector in a spherical coordinate system

In a spherical coordinate system my unit vectors are: $\vec{e_r}=\begin{pmatrix}\sin\theta\cdot \cos\phi \\ \sin\theta \cdot \sin\phi \\ \cos\theta \end{pmatrix}$; ...
0
votes
2answers
23 views

How to evaluate the gradient of a function at a point

I have a problem where I am to create a function in terms of $x$ and $y$ and compute the gradient at the point $(1,1)$. I computed the gradient but in order to evaluate it at the given point do I just ...
2
votes
1answer
43 views

Show that the set $\{x\in\mathbb{R}^N:\nabla f(x)=0 \}$ is convex

Let $f:\mathbb{R}^N\rightarrow \mathbb{R}$ be a $C^1$ convex function. Show that $\{x\in\mathbb{R}^N:\nabla f(x)=0 \}$ is convex (we assume that empty set is convex). Any hint?
2
votes
2answers
55 views

derivative of a symmetric bilinear form (quadratic form version)

Let $A=A^T\in \mathbb R^{k\times k}$ be a nonzero symmetric matrix and define $F:\mathbb R^k\to\mathbb R$ by $$f(x):=x^TAx$$ Then why $df(x)\xi=2x^TA\xi$ for $x,\xi\in\mathbb R^k$?
0
votes
0answers
22 views

Theorem proving skills in calculus, clearer idea to read in reverse order; linear-reading with writing down helps little

It is said that theorem proving skills are better trained via reproducing proofs from sketch rather than passive reading. Here we need more precise extension. e.g. in Multivariable Calculus, there ...
1
vote
3answers
38 views

Does the method for finding the intersection of 2 single variable functions work for multivariable functions?

I have $2$ multivariable functions $Q(x,y)$ and $P(x,y)$, I was wondering if finding the point of intersection between these 2 functions is as easy as making $Q(x,y) = P(x,y)$ as you would do for most ...
1
vote
1answer
38 views

Notation of multivariable derivatives

I reading a text where quadratic terms of a function $L : \mathbb{R}^2\times\mathbb{R}^2 \rightarrow \mathbb{R}^2$ are expanded as $$ L(x,\alpha)=\frac{1}{2}L_{20,0} \, x_0^2 + L_{20,1} \, x_0 x_1 + ...
1
vote
2answers
84 views

Proving that $\left( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} \right) \lvert f(z) \rvert^2 = 4 \lvert f'(z) \rvert^2$

Given $f$ entire show that $$ \left( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} \right) \lvert f(z) \rvert^2 = 4 \lvert f'(z) \rvert^2 $$ I've come close to getting the exact ...
2
votes
1answer
35 views

How to evaluate a double integral with two Dirac functions?

Here I have a problem, is the solution the same if I integrate every one? part by part? $$\int_0^Te^{-(s+\mu\lambda^2 ) t} \int_0^l\left[\delta(x-R)\delta(t-tj)\varphi(x) \, dx\, dt\right]$$ I've ...
5
votes
2answers
29 views

conical surface, parametrization, immersion, Gaussian and mean curvatures

"Find the parametric form of a conical surface $S$ which is spanned by all rays starting $($but not including $)$ a fixed point $\gamma$ and passing through an arbitrary point on $\gamma$ and passing ...
-4
votes
0answers
44 views

Surface Area in 4 dimensions [closed]

We all know the surface area relation in 3 space from multivariable calculus, however, how is this taken to 4 space?
0
votes
0answers
33 views

Create a fourth order polynomial function f(x,y) with at least two distinct terms

I will be computing the gradient, finding the critical points, and use Lagrange multipliers to either maximize or minimize the function. Any suggestions?
1
vote
1answer
20 views

Set up integral in spherical coordinates outside cylinder but inside sphere

I have the equation of a cylinder and the equation of a sphere given: Cylinder: $x^2+y^2=4$ Sphere: $x^2+y^2+z^2=25$ I'm asked to set this up in cylindrical and spherical coordinates. Cylindrical ...
4
votes
0answers
48 views

Conditions for Taylor formula

I know that, if $F:X\to Y$, where $X,Y$ are Banach spaces, is a map whose $n$-th Fréchet derivative $x\mapsto F^{(n)}(x)$ is continuous as a function of $x$ in a neighbourhood of $x_0\in X$, then the ...
0
votes
1answer
24 views

What space curves can this theorem describe?

We were given the following theorem in our Vector Calculus class: THM: For space curve $R$ which does not pass through the origin, and which has a second derivative, the following are equivalent: ...