Question - https://imgur.com/a/roEWwaw
As shown in the picture, I have done part 1 of the problem chose my own values, and found the local min and max with 2 saddle points, so in total 4 critical points. However, I don't understand how to do part 2, as far I am assuming is it asking to do the maximum directional derivative. So, all I have done so far is plug in the 4 critical points and found the maximum rate of change, however, for all 4 crit points the answer was 0. Intuitively, the answer zero makes sense because there won't be any rate of change at the local min/max/saddle points. Did I approach the problem the wrong way, is this supposed to be the answer?
In this post: https://www.physicsforums.com/threads/basis-for-tangent-space-and-cotangent-space.283108/
What is the difference between directional derivatives \partial/\partial x and differentials dx?
If the formula for directional derivatives is D = fx u1 + fy u2, are you just adding the slope in the x and y direction?
Correct me if I am wrong ,Directional Derivativ e helps us to calculate the derivative along a vector say r=ai+bj ,iand j vectors are basis vectors ,r vector is given by moving "a"units along X direction ,and "b" units along Y direction ,δf/δx gives us how much the function changes per unit length of X in i direction ,and δf/δy gives us how much the function changes per unit length of Y in j direction, since it has moved "a" units along X direction the changes the function has undergone in X direction is a(δf/δx) ,similarly in y direction it's b(δf /δy) since the changes are different in each direction shouldn't the changes be a(δf/δx )i +b (δf/δy )j ? What I am trying to say is both δf/δx ; δf /δy are along 2 different direction ,just adding them doesn't feel right,shouldn't we mention i and j vectors along with them.
Guys, where can I find physics questions which include applications of
double integrals, partial derivatives and directional derivatives
Hi, I've uploaded above a picture of my question. For some reason, I am unable to render LaTeX.
Update: I was able to solve it. Here's the link: Solution
- I know that directional derivative at a vector point a would be the dot product of the gradient of the function at a and the unit vector along the z-axis.
- I know maximum occurs when directional derivative is along the direction of the gradient.
But how do I use this to solve this question? Am I getting something wrong conceptually? What am I missing?
hello, I was wondering what I did wrong on this problem, I believed that I had solved it correctly but I'm not sure. Any help would be appreciated!!
Compute the directional derivative of 𝑓(𝑥,𝑦)=𝑒^2𝑥−5𝑦 at the point (5,2) in the direction of the point (−4,5).
I got the answer of -33/sqrt41.
Here's a picture of my work https://imgur.com/tKjckBX!
Find the directional derivative of f(x,y,z)=z3−x2y at the point (3, -4, -1) in the direction of the vector 𝐯=⟨1,1,5⟩
I have this problem's image. Please tell me how to find direction from the given line's equation. I'm comfortable with finding gradient.
I have an equation with two variables and one of the questions says to determine a 3D vector tangent to the surface in the direction of steepest ascent at the location (2,0). I know the direction of steepest ascent is the gradient so I put the values (2,0) into the gradient which <1/4,0> but I don’t know how to get a z value. I asked a tutor about this earlier and she said she wasn’t 100% sure but to try and take the magnitude of the gradient and use that. Somebody asked our teacher if that was right earlier today and she said to remember what the magnitude of the gradient is telling you about z but didn’t say anymore.
Also I have a couple other questions:
- In question a, is the magnitude of the gradient (steepness I think) the same as the gradient•(unit vector)? I got 5/8 for both of them and I’m wondering if one is redundant and why you don’t just use magnitude instead of the directional derivative or if it’s just a coincidence.
- In question b, is the slope the same as the gradient? I got the gradient at the point (1,4) and I think it’s the same but I’m not 100% sure.
Let f(x,y) be differentiable at the point (13,7) Suppose you know D<-4,-1>f(13,7)=6 And that D<-4,1>f(13,7)=-4
Determine the values of f_x(13,7) and f_y(13,7)
I’m having trouble finding an equation that sets the partial derivatives as a system of equation so I can solve. I’ve considered D<-4,-1>f(13,7)=6 = f_x(13,7)a+f_y(13,7)b is this the right approach ?
I'm struggling to understand a specific step in the process. I've attached a photo of the problem with the steps I'm talking about circled.
How do they know it's θ=π−arcsin((5√)/61)/61) and notθ=π−arccos((-6√61)/61)? And how did they get to 4√(61) for the max value from that?
Also please be nice, I was never taught trigonometry, I was just expected to learn it in high school and I never did. I had to pick it up while learning precalculus.
Thanks in advance.
Substituted carboranes with ‐SeMe, ‐TeMe, and ‐I groups on the skeletal carbon vertices are shown to be outstanding chalcogen bond (ChB) and halogen bond (XB) donors to recognize halide anions. Moreover, the interactions are found to be remarkably efficient with short distances and high directionality. Theoretical calculations further validate the presence of a deep σ‐hole on Se, Te, and I.
Abstract
Crystal engineering based on σ‐hole interactions is an emerging approach for realization of new materials with higher complexity. Neutral inorganic clusters derived from 1,2‐dicarba‐closo‐dodecaborane, substituted with ‐SeMe, ‐TeMe, and ‐I moieties on both skeletal carbon vertices are experimentally demonstrated herein as outstanding chalcogen‐ and halogen‐bond donors. In particular, these new molecules strongly interact with halide anions in the solid‐state. The halide ions are coordinated by one or two donor groups (μ1‐ and μ2‐coordinations), to stabilize a discrete monomer or dimer motifs to 1D supramolecular zig‐zag chains. Crucially, the observed chalcogen bond and halogen bond interactions feature remarkably short distances and high directionality. Electrostatic potential calculations further demonstrate the efficiency of the carborane derivatives, with Vs,max being similar or even superior to that of reference organic halogen‐bond donors, such as iodopentafluorobenzene.
https://ift.tt/3c1kYcv
I need to learn everything I can about the gradient and directional derivatives before my exam tomorrow morning. Any quick readings, exercises or pro tips would be greatly appreciated. Cheers
Can we think of directional derivatives as simply being the component of the gradient of f onto unit vector u (or v or whatever) as we are taking the dot product of the gradient of f and u? Or am I seeing a connection that isn't there?
I understand & can visualize what a directional derivative is - basically the slope at a particular point on a graph when your pointing in the direction of an arbitrary vector, & I understand that the directional derivative (slope) changes depending on what direction your facing, just like if you were on a mountain side.
What I don't understand & am struggling to visualize is - why is the directional derivative equal to the gradient dotted w/ your particular unit vector? I can't visualize what that looks like, and I've looked all over the web without seeing any answer that is satisfying to me.
I'm tring to picture a bunch of vectors pointing in every direction at a particular point (arrows in every direction sort of forming a circle), and I understand one of them is the gradient vector pointing in the direction of greatest descent/ascent - but why would dotting that gradient vector w/ any of the others give you the magnitude of the slope at that point in the direction of said vector?
Does anyone have a visualization showing the scalar projection of the unit vector onto the gradient (I guess that's ultimately what I'm looking for)?
The directional derivative of f with respect to some vector v and -v gives the same absolute value but opposite signs. Is the opposite sign because ratio in the limit definition is (f(x-hv)-f(x))/h vs. (f(x+hv)-f(x))/h, and if we are interpreting this as the slope of the tangent line, for some “run” magnitude, the “rise” has opposite signs since we are stepping in opposite directions?
In the regular and partial derivatives, I don’t think this issue exists. Is the reason because it’s built into the definition that we are stepping in the positive direction along the axis?
From Khan Academy, the directional derivative is defined as D_u f(x_0) = lim (f(x_0 + hu) – f(x_0))/h as h-> 0 (here x_0 and u are vectors).
I have two calculus books that define u as a unit vector, is the reason because the limit definition above become analogous to the regular and partial derivatives? I’m imagining the ratio in the limit as basically evaluating f(x) at x_0 and h away from x_0 in the direction of u, then dividing by h, so as h -> 0, the limit is basically giving us the slope of the tangent line at x_0 if we section f(x) along u?
How does one interpret the directional derivative when u is not a unit vector?
Hi guys, I want to know how to take the directional derivative for a linear mapping/matrix, however I cant seem to find tutorials online that show me how to do so. Could someone tell me if the term I am searching for is wrong or point me to some tutorials that illustrate how to do this? Thanks in advanced!
So i was asked if the function (top right) was differentiable in (x0,y0)=(0,0).
I just said that it isn‘t because you can‘t integrate |x| for x=0. (Which you would need for the formula in the bottom.
But the solutions said that it is and with the formula in the middle you get the solution f‘(0,0)=(0,0).
So where did i go wrong, can‘t i assume f(x0, y0) can‘t be differentiate because it can‘t be partially differentiate after one of the variables for given (x0, y0)? And how would i solve such a question if it‘s asked for a certain direction (with vector v), because i can‘t use the bottom formula i always use.
https://youtu.be/N_ZRcLheNv0 In this video, Grant asserts that, a small nudge in a direction defined by the vector v= [a b], hv (h is infinitesimally small) is made of two small nudges ha and hb along x and y directions, and immediately concludes that the derivative along the direction v must be the sum of "a" times the partial of "f" wrt x and "b" times the partial if f wrt y. Why is the latter true ?
Dear people of /r/learnmath
I've been struggling with this question for quite some time and I was hoping you guys could help me understand this question. The original question was (I wrote it in latex in the hope everyone can understand):\\
"Consider a function $f:\mathbf{R}^2 \to \mathbf{R}$ with $\bigtriangledown f=x^2yi+(1-2y)j$. Determine the directional derivative of $f$ in point $P(\sqrt{3},1)$, in the direction of $-4i+3j$." \\
To solve this I first stated that $\bigtriangledown f(x,y)=(\frac{dz}{dx},\frac{dz}{dy})=(2xyi,x^2i-2j)$. Filling in $P(\sqrt{3},1)$ gives $\bigtriangledown f(\sqrt{3},1)=(2\sqrt{3}i,3i-2j)$. Taking the direction $||\bigtriangledown f(\sqrt{3},1)||=\sqrt{(2\sqrt{3}i)^2+(3i-2j)^2}=\sqrt{21i^2-12ij+4j^2}.\\
After this I get fully stuck, I have no idea how to implement the $-4i+3j$. This is a question of a practice exam, but due to corona they said they will ask the question differently "Replace the question as follows: Determine in which unit direction the function has maximal rate of change".\\
So I looked up information on the internet and found this site: http://tutorial.math.lamar.edu/Classes/CalcIII/DirectionalDeriv.aspx , Here they state something with the angle being equal to $\pi$ and the they gradients are equal so the so a maximal rate of change. This makes sense, but I don't know how to implement this. Could someone explain how to do this question and implement the $-4i+3j$. This would make my day. Kind regards
I was hoping that someone could intuitively explain directional derivatives or direct me to a video that helped them understand it
Hi guys, I'm currently taking an intro to deep learning course at my university, however I got totally lost when the instructor started going through directional derivatives of linear maps. I'm not entirely sure if that is the correct term I'm supposed to be searching for because I'm not getting any useful results from google. I understand directional derivatives for multivariable functions, however, I just cant carry that intuition over to higher dimensional cases like matrices.
Below are some of the questions. Could someone point me to some tutorials or articles that explain these concepts? Thanks in advance!
https://preview.redd.it/bo434yivyuf41.png?width=513&format=png&auto=webp&s=b5651d781db14a283c171b8102e77da2e2fdce59
Correct me if I am wrong ,Directional Derivativ e helps us to calculate the derivative along a vector say r=ai+bj ,iand j vectors are basis vectors ,r vector is given by moving a units along X direction ,and b units along Y direction ,δf/δx gives us how much the function changes per unit length of X ,and δf/δy gives us how much the function changes per unit length of Y, since it has moved a units along X direction the changes the function has undergone in X direction is a(δf/δx) ,similarly in y direction it's b(δf /δy) since the changes are different in each direction shouldn't the changes be a(δf/δx )i +b (δf/δy )j ? What I am trying to say is both δf/ δx ; δf /δy are along 2 different direction ,just adding them doesn't feel right,shouldn't we specify the direction in which the change is happening ,that is how much does the surface change along X direction ,then how much it changes along Y direction .
Just checkin
Substituted carboranes with ‐SeMe, ‐TeMe, and ‐I groups on the skeletal carbon vertices are shown to be outstanding chalcogen bond (ChB) and halogen bond (XB) donors to recognize halide anions. Moreover, the interactions are found to be remarkably efficient with short distances and high directionality. Theoretical calculations further validate the presence of a deep σ‐hole on Se, Te, and I.
Abstract
Crystal engineering based on σ‐hole interactions is an emerging approach for realization of new materials with higher complexity. Neutral inorganic clusters derived from 1,2‐dicarba‐closo‐dodecaborane, substituted with ‐SeMe, ‐TeMe, and ‐I moieties on both skeletal carbon vertices are experimentally demonstrated herein as outstanding chalcogen‐ and halogen‐bond donors. In particular, these new molecules strongly interact with halide anions in the solid‐state. The halide ions are coordinated by one or two donor groups (μ1‐ and μ2‐coordinations), to stabilize a discrete monomer or dimer motifs to 1D supramolecular zig‐zag chains. Crucially, the observed chalcogen bond and halogen bond interactions feature remarkably short distances and high directionality. Electrostatic potential calculations further demonstrate the efficiency of the carborane derivatives, with Vs,max being similar or even superior to that of reference organic halogen‐bond donors, such as iodopentafluorobenzene.
https://ift.tt/3c1kYcv
Crystal engineering based on σ ‐hole interactions is an emerging approach for realization of new materials with higher complexity. Neutral inorganic clusters derived from 1,2‐dicarba‐ closo ‐dodecaborane, substituted with –SeMe, –TeMe, and –I moieties on both skeletal carbon vertices are experimentally demonstrated here as outstanding chalcogen‐ and halogen‐bond donors. In particular, these new molecules strongly interact with halide anions in the solid‐state. The halide ions are coordinated by one or two donor groups ( μ 1 ‐ and μ 2 ‐coordinations), to stabilize a discrete monomer or dimer motifs to 1D supramolecular zig‐zag chains. Crucially, the observed chalcogen bond and halogen bond interactions feature remarkably short distances and strong directionality. Electrostatic potential calculations further demonstrate the efficiency of the carborane derivatives, with V s,max being similar or even superior to that of reference organic halogen bond donors such as iodopentafluorobenzene.
https://ift.tt/3c1kYcv
