# Tagged Questions

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### A generalization of the mean value theorem?

Let $U \subset \mathbb{R}^d$ be open and path-connected. Let $f: U \to \mathbb{R}^m$ be differentiable on $U$ and suppose there exists a real $M$ such that $|| D_f(x) || \leq M$ for all $x \in U$. ...
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### Difference between Vector Functions and Vector Field

I understand that a vector function is a function that has a domain $\mathbb{R}^n$ and range on $\mathbb{R}^m$ so it takes vectors and gives vectors right? So what is a vector field?And how can I ...
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### Continuous functions. Second norm

Let $f:[a,b]\rightarrow \mathbb{R}^{d}$ be continuous. I need to prove that $\left \| \int_{a}^{b}f(x)dx \right \|_{2}\leq \int_{a}^{b}\left \| f(x)) \right \|_{2}dx$.
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### Solve for the tangent plane using the gradient

I am having a hard time finishing this problem up: Consider the surface $4 x^{2} + 9 y^{2} + 4 z^{2} = 17$ and the point $P = \left( 1, 1, 1 \right)$ on this surface. A) Find the outward unit ...
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### Directional Derivatives using Polar Coordinates

I am having a hard time with this problem on my homework assignment. Here is the problem, and i will show my work below: If $f( x, y) = -2 x^{2} + 3 y^{2}$, find the value of the directional ...
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### Product rule type formula for $\nabla \cdot (M(x)v(x))$ where $M(x)$ is a matrix and $v(x)$ is a vector?

Let $M(x)$ be a $n\times n$ matrix with each element depending on $x$ a variable on $\mathbb{R}^n$. Let $v(x)$ be a vector. Is there a nice product rule formula for $\nabla \cdot (M(x)v(x))$?
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### How to find the value of tangent vectors?

In the figure $z_1=a_1+ib_1$ and $z_2=a_2+ib_2$ are the two curves. $T_1$ and $T_2$ be the tangents on the curves $z_1$ and $z_2$. What I am interested to know what will be the tangent vectors?
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Let $f: \mathbb{R}^n \to \mathbb{R}$ be a differentiable function. One way to define the gradient of $f$ is as the vector whose inner product with any other vector gives the directional derivative in ...
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### How to find $\int_{S^2}f \cdot n \ \text{d}S$ if $f(x,y,z):=(x^3,y^3,z^3)^T$

With $\mathbb{S}^2$ being the unit sphere, how to find $$\int\limits_{\mathbb{S}^2} \vec{f} \cdot \vec{n} \ \text{d}S$$ if $\vec{f}(x,y,z):=(x^3,y^3,z^3)^T$? Apparently, we need to use Gauss. ...
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### Problems with the integration law of Gauss.

I'm havin problems understanding the integration law by Gauss which states: $$\iiint\limits_{G} \operatorname{div}(\vec{w})\, dV = \iint\limits_{\partial G} \vec{w} \cdot \vec{n } \, dA$$ (I don't ...
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### What does $\Delta$ mean in context of vector calculus?

I'm reading an article that has a formula for $\Delta \phi(x)$, where $\phi : \mathbf{R}^2 \rightarrow \mathbf{R}$ and $x \in \mathbf{R}^2$ and $\Delta \phi(x) : \mathbf{R}^2 \rightarrow \mathbf{R}$ ...
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### Calculation of a curvilinear integral

Please help to calculate the following integral. TCalculate $$\int_\gamma \frac{x\,dx + y\,dy+z\,dz}{x^2+y^2+z^2}$$ where $\gamma$ is the way of class $\mathcal C^1$ which unites point on the ...
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### Proof of Gauss' Law of gravitation without reference to Newton?

Gauss' Law of gravity is: $$\bigtriangledown \cdot \mathbf{g}= 4\pi G\rho$$ This can be shown to be equivalent to Newton's Law of gravity via the divergence theorem. However, this does not really ...
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### Multivariable Calculus - Calculating Derivative Matrix

I'm working with Munkres' Analysis on Manifolds. From chapter 2 (this isn't a homework question): Given $f: \mathbb R^2 \rightarrow \mathbb R^2 : f(r,\theta)=(r\cos(\theta),r\sin(\theta))$, ...
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### Non-divergence form of a 2nd order PDE

This might be a trivial question but I'm very rusty with regards to calculus and am new to PDEs. How would you write the following second order quasilinear equation in it's non-divergence form: The ...
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### Mapping Confusion -Implicit Function Theorem-

Here is the Implicit Function Theorem statement: "Let $g : R^k \times R^n \to R^n$ be a continously differentiable function s.t. $g(x_0, y_0) = c$ and $D_yg(x_0,y_0) : R^n \to R^n$ is an isomorphism. ...
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### Find the Unit Normal Vector - Calc III

For the curve given by: $r(t)= [\sin(t) - t\cos(t), \cos(t) + t\sin(t), 6t^2 + 2]$ Solve for the Unit Normal Vector N(t). I was successfully able to solve the Unit Tangent Vector $T(t)$ as ...
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### Find the Vector Equation of a line perpendicular to the plane.

Question: Find the vector equation $r(t)$ for the line through the point $P = (-1, -5, 2)$ that is perpendicular to the plane $1 x - 5 y + 1 z = 1$. Use $t$ as your variable, $t = 0$ should ...
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### Irrotational implies path independent

I wanted to prove that if $F$ is differentiable and irrotational then it's path independent. The plan of actions is straightforward: let $C_1$ and $C_2$ be two curves, then $C$ is a closed curve then ...
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### What if $\operatorname{div}f=0$?

Say, we have a function $f\in C^1(\mathbb R^2, \mathbb R^2)$ such that $\operatorname{div}f=0$. According to the divergence theorem the flux through the boundary surface of any solid region equals ...
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### Finding the sphere surface area using the divergence theorem and sphere volume

The divergence theorem allows us to go between surface and volume (in some sense), so a natural example would be to compute the surface area of the unit sphere $U$ assuming we know the sphere volume. ...
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### Determine the resultant for the vector sum: 10N at 045° and then 8 N at 068°?

So if you draw it out on a quadrant the angle between 10N and 8N should be 68 - 45 = 23° right? So can i do cosine law to solve the resultant? c^2 = 10^2 + 8^2 - 2(10)(8)cos 23 c = 4.09 is this ...
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### How to find the distance between two planes?

The following show you the whole question. Find the distance d bewteen two planes \begin{eqnarray} \\C1:x+y+2z=4 \space \space~~~ \text{and}~~~ \space \space C2:3x+3y+6z=18.\\ \end{eqnarray} ...
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On the one hand there is an explicit formula for divergence in spherical coordinates, namely: $$\nabla \cdot \vec{F} = \frac{1}{r^2} \partial_r (r^2 F^r) + \frac{1}{r \sin \theta} \partial_\theta ... 5answers 241 views ### What is the intuition behind the unit normal vector being the derivative of the unit tangent vector? I've seen the math, but... It just doesn't make sense to me. How is the slope going to point perpendicular to the vector that is clearly a straight line not going in that direction? 1answer 35 views ### Solving surface integral I just need some help on solving surface integral. Actually I already finished doing part (a) and part (b) but just part (c) I dont know how to do it. It would be nice if someone able to guide me to ... 2answers 91 views ### Verifying Stokes Theorem on Vector Calculus My question is to verify Stokes Theorem. I manage to do using Stokes Theorem \iint_R \nabla\times\vec{F}\cdot d\textbf{S} and got my answer 7/6 but I dont know how to do the direct line ... 1answer 37 views ### Simplify vector equation I know that div E=0 and I know what  curl E is. Further, I know what the vector laplacian of E is. Now I want to simplify \nabla \times (\nabla \times f(x,y,z) E(x,y,z)), where ... 1answer 94 views ### Yang–Mills theory We define the energy as$$E = I_F + I_K + I_V,$$where,$$I_F [A]= \frac{1}{2} \int d^Dx \operatorname{tr} F^2_{ij},$$F_{ij} represents the electromagnetic force.$$I_K [\phi,A]= \frac{1}{2} \int ...
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Example 3: Sketch the gradient vector field for $f(x,y) = x^2 + y^2$ as well as several contours for this function. The gradient of the vector field is $$\nabla f(x,y)=2x\vec{i}+2y\vec{j}.$$ But ...
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### Trouble understanding a common vector calculus example

I have difficulty understanding the following vector calculus example. Text can be found here. It is the 5th Q&A -- starting with equation (31.1035).It concerns finding the vector potential of a ...
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### Can I treat vectors in $\Bbb R^3$ with a component equal to zero as vectors in $\Bbb R^2$?

If I have a vector $(0,4,4)$ and was finding perpendicular unit vectors to this, is it the same case as finding perpendicular unit vectors for the vector $(4,4)$? Meaning a maximum of two possible ...