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

3
votes
2answers
22 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
3answers
34 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: ...
0
votes
0answers
17 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
23 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
49 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
26 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
38 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
25 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
19 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
29 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
32 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
23 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
36 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
43 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.
10
votes
3answers
229 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
41 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
54 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
37 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
28 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 [on hold]

We all know the surface area relation in 3 space from multivariable calculus, however, how is this taken to 4 space?
0
votes
0answers
32 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
23 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: ...
0
votes
2answers
35 views

Is this function continuous? (vector function)

Assume you have $k$ vectors: $\{v_1,\dots,v_k\}$ in $\mathbb{R}^n$, and $\lambda\in\mathbb{R}^k$. Look at the function: $F\colon\mathbb{R}^k\rightarrow \mathbb{R}^n$ where ...
3
votes
1answer
36 views

Using Stokes' Theorem to evaluate $\displaystyle\int_{C}{(xyz)dx+(xy)dy+(x)dz}$

Let $C$ be the closed, piecewise smooth curve formed by traveling in straight lines between the points $(0,0,0),(2,1,5),(1,1,3)$ and back to the origin, in that order. Use Stokes' theorem to evaluate ...
2
votes
0answers
38 views

Spaces of the derivative in a direction

I have two question regarding the spaces where the first, and second, directional derivatives of a functional are. Let $\Omega \in \mathbb{R}^2$ an open subset. Let the functional: $$\phi =L^p ...
0
votes
1answer
39 views

Why do the limits of integration matter in a double integral?

Okay, I know that seems like a stupid question but I couldn't think of a better way to phrase it. I was trying to understand why iterated integrals involve "projecting" the domain onto one of the ...
0
votes
1answer
58 views

Show that $f$ is a linear map if $f$ is differentiable and its derivative is constant:

Show that if $f:ℝ^m→ℝ^n$ is a differentiable function whose derivative function $f′$ is a constant function and such that $f(0)=0$, then $f$ is a is a linear map. I am a little lost on this. I know ...
5
votes
1answer
186 views

Riemann Integrability in $\Bbb R^2$

Define the General Subdivision $S$ of a rectangle $R$ in $\Bbb R^2$ as a collection $E_1,...,E_k$ of Jordan regions such that none of them has interior points in common, and: $$R \subset ...
2
votes
0answers
12 views

Finding the Area of a Torus-like surface

I'm trying to find out the Area of the following surface: Let $C$ be the curve associated to a regular, simple path $\theta:[0,l]\rightarrow \Bbb R^2 $; also assume that ...
2
votes
1answer
18 views

A question regarding Surface Integrals and Stoke's Theorem

Let $G$ be an open set in $ \Bbb R^3$ and $F:G \rightarrow \Bbb R^3-{0}$ a vectorial field of class $C^1$. Suppose that $S$ is an open set, contained in $G$, whose non-empty boundary $\delta S$, is ...
1
vote
1answer
19 views

Calculating Flux through surface, stokes theorem, cant figure out parameterization of vector field

I am a tutor and trying to solve this for a student. I understand that Stokes's theorem allows us to compute the flux through the surface S, instead through the surface of the unit disk because they ...
0
votes
0answers
19 views

Finding the volume between a cone and a sphere

I have to find the volume between the sphere $x^2+y^2+z^2=1$ and below the cone $z=\sqrt{x^2+y^2}$ using Spherical Coordinates. Here is what I have so far: Transforming the cone part gives: ...
0
votes
0answers
20 views

mistake by computing derivatives

Let be $f: \mathbb{R}^{2} \to \mathbb{R}^{2},(x_1 , x_2) \mapsto (f_1 (x_1 , x_2), f_2 (x_1 , x_2)) $ and $d: \mathbb{R} \to \mathbb{R}^2, t \to (d_1(t), d_2(t))$ be smooth functions. Define $c:= ...
1
vote
0answers
19 views

Derivative with respect to r in terms of Cartesian coordinates?

I am trying to show that $$\sum_{j=1}^3 x_j \frac{\partial}{\partial x_j} = r \frac{\partial}{\partial r}.$$ I have $$\frac{\partial f}{\partial x_j} = \frac{\partial f}{\partial r} \frac{\partial ...
3
votes
2answers
78 views

What is difference between all of these derivatives?

In calculus II we were introduced to a bunch of new derivatives: the gradient, the derivative $D=\begin{bmatrix} \partial_{x_1} \\ \partial_{x_2} \\ \vdots \\ \partial_{x_n}\end{bmatrix}$, the ...