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Polar Transform

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[x,y] = pol2cart(theta,rho) transforms corresponding elements of the polar coordinate arrays theta and rho to two-dimensional Cartesian, or xy, coordinates. example [ x , y , z ] = pol2cart( theta , rho , z ) transforms corresponding elements of the cylindrical coordinate arrays theta , rho , and z to three-dimensional Cartesian, or xyz , coordinates Polar and Log-Polar Transformations. Rotation differences between two images can be converted to translation differences along the angular coordinate ( θ) axis of the polar-transformed images. Scaling differences can be converted to translation differences along the radial coordinate ( ρ) axis if it is first log transformed (i.e., ρ = ln. ⁡ Lecture L3 - Vectors, Matrices and Coordinate Transformations By using vectors and defining appropriate operations between them, physical laws can often be written in a simple form. Since we will making extensive use of vectors in Dynamics, we will summarize some of their important properties. Vector Thus, in terms of polar coordinates, the Fourier transform operation. transforms the spatial position radius and angle ( r, θ) to the frequency. radius and angle (ρ, ψ ). The usual polar. In many practical situations, it will be necessary to transform the vectors expressed in polar coordinates to cartesian coordinates and vice versa. Since we are dealing with free vectors, we can translate the polar reference frame for a given point (r,θ), to the origin, and apply a standard change of basis procedure

Polar Transformer - National Institutes of Healt

Log-polar transform

polarTransform · PyP

  1. Writing the frequency φ ={rcos(θ),rsin(θ)} in Polar coordinates, we let f(r,θ)˜ =fˆ φ(r,θ). In this paper, the term Polar Fourier transform will always refer to the operation f(r,θ)˜ =PF f(x), namely, getting f(x)in Cartesian variables and computing f(r,θ)˜ defined with Polar variables
  2. Polar transform in CSS3? Ask Question Asked 9 years, 5 months ago. Active 2 years, 1 month ago. Viewed 2k times 13 2. Turning a line into a ring is a simple task in graphics programs such as GIMP: (source: adamhaskell.net) I'm trying to work out.
  3. Learn how to convert from polar to rectangular coordinates in this free math video tutorial by Mario's Math Tutoring.0:11 Overview of Polar and Rectangular C..
  4. g Polar to Cartesian andVice Versa In discussion section I mentioned that once we specify an origin, we have a one-to-one correspondance between vectors and points. Given a point, we just associate the vector which starts at the origin and points to that point. Goin
  5. Laplace's equation in the Polar Coordinate System As I mentioned in my lecture, if you want to solve a partial differential equa-tion (PDE) on the domain whose shape is a 2D disk, it is much more convenient to represent the solution in terms of the polar coordinate system than in terms of the usual Cartesian coordinate system

Using Polar Coordinates we mark a point by how far away, and what angle it is: Converting. To convert from one to the other we will use this triangle: To Convert from Cartesian to Polar. When we know a point in Cartesian Coordinates (x,y) and we want it in Polar Coordinates (r,.

Polar coordinate system - Wikipedi

  1. We introduce a novel adaptive polar transform (APT) technique that evenly and effectively samples the image in the Cartesian coordinates
  2. [theta,rho] = cart2pol(x,y) transforms corresponding elements of the two-dimensional Cartesian coordinate arrays x and y into polar coordinates theta and rho
  3. Ett stort urval av Polar Utrustning online på Sportamore. Sportvarumärken för henne. 365 dagar att returnera och alla produkter finns i lager
  4. To transform the polar point ( 4, 3 π 4) into rectangular coordinates: first identify (r, θ). r = 4 and θ = 3 π 4. Second, draw a vertical line from the point to the polar axis (the horizontal axis). The distance from the pole to where the line you just drew intersects the polar axis is the x value, and the length of the line segment from.
  5. The other one works for points expressed in polar coordinates. 2) The transformation from cartesians to polars is not linear because if it were, then it wouldn't make sense to transform problems because the arithmetic properties would remain the same

How to transform from rectangular coordinates to polar coordinates? We remember that the rectangular coordinates are written in the form and the polar coordinates are written in the form , where r is the distance from the origin to the point and θ is the angle formed between the line and the x-axis.These coordinates are related using trigonometry In this section we will introduce polar coordinates an alternative coordinate system to the 'normal' Cartesian/Rectangular coordinate system. We will derive formulas to convert between polar and Cartesian coordinate systems. We will also look at many of the standard polar graphs as well as circles and some equations of lines in terms of polar coordinates Polar transform makes the algorithm easily and more accurately to find rotation angle under the fact that rotation in Cartesian domain corresponds to pure translation in polar domain. And the widely used sum of absolute difference (SAD) matching criteria and modified gradient descent search algorithm are introduced to improve the search speed on the condition that does not decrease the search. matplotlib.projections.polar ¶ class matplotlib.projections.polar.InvertedPolarTransform (axis = None, use_rmin = True, _apply_theta_transforms = True) [source] ¶. Bases: matplotlib.transforms.Transform The inverse of the polar transform, mapping Cartesian coordinate space x and y back to theta and r Ebene Polarkoordinaten (mit Winkelangaben in Grad) und ihre Transformation in kartesische Koordinaten. Die Koordinate , eine Länge, wird als Radius (in der Praxis auch als Abstand) und die Koordinate als (Polar)winkel oder, in der Praxis (gelegentlich) auch als Azimut bezeichnet.. In der Mathematik wird meistens der Winkel im Gegenuhrzeigersinn als positiv definiert, wenn man senkrecht von.

The Log-Polar transform diagram

Transform coordinates Online convertor for lat & long coordinates, geodetic datums and projected systems. Input coordinate system Not selected Change Input coordinates Batch. X: Y: Show position on a map. More details. Output coordinate system Not selected Change Output coordinates X: Y: Show. In polar coordinate space, the meaning of r describes the distance of a point (x,y) to the origin position in Cartesian coordinate space, and ϕ discribes the angle of vector and its range is from 0 to 359°. Due to the origin symmetric of polar coordinate transform, the transform needs to be carried out in the range of 0° to 179° Polar to Rectangular Online Calculator. Below is an interactive calculator that allows you to easily convert complex numbers in polar form to rectangular form, and vice-versa. There's also a graph which shows you the meaning of what you've found. For background information on what's going on, and more explanation, see the previous pages Polar mapping can be linear or semi-log. Add one of WarpPolarMode to flags to specify the polar mapping mode. Linear is the default mode. The semilog mapping emulates the human foveal vision that permit very high acuity on the line of sight (central vision) in contrast to peripheral vision where acuity is minor. Option on dsize

Video: Introduction to Polar Codes: Polar Transform - YouTub

Solved: polar transformation - PTC Communit

Fast Polar Fourier Transform FFT is one of top 10 algorithms of 20th century. We'll extend utility of FFT algorithms to new class of settings in image processing. Create a tool which is: Free of emotional involvement, & Freely available over the internet. Current Stage In this section we will look at converting integrals (including dA) in Cartesian coordinates into Polar coordinates. The regions of integration in these cases will be all or portions of disks or rings and so we will also need to convert the original Cartesian limits for these regions into Polar coordinates Finally, we describe an intrinsic connection between the polar Fourier transform and the filtered backprojection method that has recently been introduced to process projection-reconstruction NOESY data. Direct polar Fourier transformation holds great potential for producing quantitatively accurate spectra from radially sampled NMR data Probability Density Under Transformation Pramook Khungurn September 25, 2015 1 Introduction In creating an algorithm that samples points from some domain, a problem that always comes up is the following: Let Aand Bbe sets, p A() be a probability density on A, and fbe a function from Ato B. If one samples xfrom Aaccording to

Fast and Accurate Polar Fourier Transform A. Averbuch⁄ R.R. Coifmany D.L. Donohoz M. Eladx M. Israeli{May 22nd, 2005 - Revised Version Abstract In a wide range of applied problems of 2-D and 3-D imaging a continuous formulation of the problem places great emphasis on obtaining and manipulating the Fourier transform in Polar coordinates. How The transformation from Cartesian to polar coordinates is not a linear function, so it cannot be achieved by means of a matrix multiplication. Share. Cite. Follow answered Jun 24 '12 at 12:01. Jyrki Lahtonen Jyrki Lahtonen. 121k 19 19 gold badges 235 235 silver badges 561 561 bronze badge Polar To/From Rectangular Transform of Images. version 1.0.0.0 (47 KB) by Prakash Manandhar. converts rectangular image to polar and back. 4.3 (26) 9.9K Downloads. Updated 17 Dec 2007. View License. × License. Follow; Download. Overview. I read various papers about the log polar transform and its application on template matching with images and have some questions: In Image Registration Using Log Polar Transform and Phase Correlation to Recover Higher Scale the authors say: In this algorithm first the sense image is downscaled by the factor of 2

You can't use the log-polar transform on its own to match images where there is translation as well as scale and rotation change. One approach is to use a Fourier spectrum representation which is translation-independent, and use the log-polar representation of this to deal with the scale and rotation Thus, log-polar transformations were repeated for c ∈ {2, 4, 8} cores. Results for 16 cores were also conducted due to the Hyperthreading capabilities of the Intel processor, with no extra performance benefit. Speed-up results are shown in Fig. 8. The log-polar transform, as implemented, is a memory-bound algorithm ROBUST IMAGE REGISTRATION USING LOG-POLAR TRANSFORM George Wolberg Siavash Zokai Department of Computer Science City College of New York New York, NY 10031 wolberg|zokai @cs-mail.engr.ccny.cuny.edu ABSTRACT yields a robust solution that precisely registers images with This paper describes a hierarchical image registration algo- subpixel accuracy. rithm for affine motion recovery Coordinate Transformations The field of mathematics known as topology describes space in a very general sort of way. Many spaces are exotic and have no counterpart in the physical world. Indeed, in the hierarchy of spaces defined within topology, those that can be described by a coordinate system are among the more sophisticated

Detailed Description. The functions in this section perform various geometrical transformations of 2D images. They do not change the image content but deform the pixel grid and map this deformed grid to the destination image. In fact, to avoid sampling artifacts, the mapping is done in the reverse order, from destination to the source polarTransform is a Python package for converting images between the polar and Cartesian domain. It contains many features such as specifying the start/stop radius and angle, interpolation order (bicubic, linear, nearest, etc), and much more. 1.2License polarTransform has an MIT-basedlicense. 1.3Installing 1.3.1Prerequisites •Python 3. (c) log-polar transform of (a) (d) log-polar transform of (b) Figure 2: Log-polar coordinate transformation. assumed that the originof both images lie at their geometric centers. In fact, their centers can be displaced and unless correspondence (translation) is known, the information deri ved from polar transformation is limited alue 2D FOURIER TRANSFORMS IN POLAR COORDINATES Natalie Baddour Department of Mechanical Engineering, University of Ottawa, 161Louis Pasteur, Ottawa, Ontario, K1N 6N5, Canada Email: nbaddour@uottawa.c

Polar Transformations. 35 likes. Just For Fun. See more of Polar Transformations on Faceboo The theory of the continuous two-dimensional (2D) Fourier Transform in polar coordinates has been recently developed but no discrete counterpart exists to date. In the first part of this two-paper series, we proposed and evaluated the theory of the 2D Discrete Fourier Transform (DFT) in polar coordinates. The theory of the actual manipulated quantities was shown, including the standard set of. polar transform cartesian conversion logPolar linearPolar cv2 opencv radius theta angle, cartesian, conversion, image-processing, polar, python, python3 License MIT Install pip install polar-transform==1.

Polar Transform学习记录_PyBigStar的博客-CSDN博

  1. 2340 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 18, NO. 10, OCTOBER 2009 Image Registration Using Adaptive Polar Transform Rittavee Matungka,StudentMember,IEEE, YuanF.Zheng,Fellow,IEEE,and RobertL.Ewing,SeniorMember,IEEE Abstract—Image registration is an essential step in many image processing applications that need visual information from mul
  2. 458 COORDINATE TRANSFORMATIONS which can be represented in matrix form as Polar celestial coordinates are right ascension (the celestial analog of longitude, measured eastward from the vernal equinox) and declination (the celestial analog of latitude), as illustrated in Fig. C.3
  3. Polar harmonic transforms (PHTs) are a kind of continuous orthogonal moments (COMs) that have excellent image description capability. To improve the anti-noise and reconstruction performance of PHTs, integer-order PHTs are generalized to order by a set of novel decimal-order polar harmonic transforms (DPHTs)
  4. So far, we have seen that part of the Fourier transforms merely wraps a time-series signal in polar coordinates. But how does this help us identify what sinewaves are used to make up the green signal? The frequency term in the exponent can be thought of as the frequency at which we are wrapping the time-series signal in the polar plane
  5. For improving the segmentation performance further, we also introduce the polar transformation, which provides the representation of the original image in the polar coordinate system. The experiments show that our M-Net system achieves state-of-the-art OD and OC segmentation result on ORIGA dataset
Schematic of the quasi-polar transform

transformation equations transforming from [ ]T u 1 u 2 to [ ] T u 1 u 2′: 11 22 cos sin sin cos u u u u θθ θθ ′ ′ = − (1.5.4) An important property of the transformation matrix is that it is , by which is orthogonal meant that [−1]=[Q T] Orthogonality of Transformation/Rotation Matrix (1. 5 Polar-coded modulation is built by the cascaded two-step channel transform. At step 1, the underlying channel W with input 2 m -ary symbols is partitioned into an ordered set of m bit synthesized subchannels under SP, each followed by a conventional N -dimension CP in step 2, finally resulting a series of m N correlated bit polarized subchannels 3)Moreover, a polar transformation is utilized in our method to transfer the fundus image into a polar coordi-nate system, which introduces the advantages of spatial constraint, equivalent augmentation, and balancing cup proportion, and improves the segmentation performance. 4)At last, we evaluate the effectiveness and generalizatio

Convert between Polar and Cartesian Coordinate

  1. Polar Claw (Mega, 1996) Takara ID number: C-2; Accessories: Batscout claw/bat partner Part of the very first Beast Wars Mega assortment, Polar Claw transforms into a polar bear. Rearing him up on his hind legs in beast mode exposes a pair of nasty-looking fangs
  2. I have been going through the latest OpenCV implementation of the Log Polar Transform using the Blind Spot Model. I am a bit confused as to how the cortical image size (sectors x rings) is being calculated. The inputs I provide are width & height of original image, center point, rings, blindspot size and I ask OpenCV to internally calculate the sectors
  3. Transformations can additionally be given as CoordinateTransformData standard names { oldsys -> newsys, metric, dim }, where { oldsys, metric, dim } and { newsys, metric, dim } are valid charts available from CoordinateChartData. The short form in which dimension is omitted may be used. CoordinateTransform [ t, pt] is effectively equivalent to.
  4. EE301 - PHASORS, COMPLEX NUMBERS IN AC AND IMPEDANCE 2 9/22/2016 Geometric Representation We represent complex numbers geometrically in two different forms. In the rectangular form, the x-axis serves as the real axis and the y-axis serves as the imaginary axis. So, for example, the complex number C = 6 + j8 can be plotted in rectangular form as
  5. Polar harmonic transform In [40] , Yap et al. introduced a two-dimensional transformation named PHT based on a set of orthogonal projection bases. Compared with other orthogonal moments, PHT achieves better performance in image reconstruction, lower noise sensitivity and lower computational complexity
  6. Nonlinear Polar Transformer. Description: The plugin performs parametric angular and radial polynomial transformations. It corrects X-ray image radial and angular distortions originated in image intensifiers of the type with electron accellerators. It allows for center-symmetrical radial and angular transformation of an image e.g. the radius r.
  7. The easiest way to remember the polar coordinate formulas is in terms of the area di erential dA. For rectangular coordinates, dA= dxdy. But in po-lar coordinates, dA= rdrd . That's because the Jacobian of the transformation is just r. Polar coordinates. The equations to convert between rectangular and polar coordinates are x= rcos r 2= x2 +

Complex Numbers and Phasors in Polar or Rectangular For

  1. Polar + Rectangular transformations Plugin What's this? This is a plugin under Effect/Distort menu. It should replace Polar to Rect and Rect to Polar, as I'm merging the two in one. Giving tribute to the original authors, @Illnab1024 for Polar to Rect Conversion Plugin, @Programmerman for Rectang..
  2. Spherical-polar coordinates . 1.1 Specifying points in spherical-polar coordinate s . To specify points in space using spherical-polar coordinates, we first choose two convenient, mutually perpendicular reference directions (i and k in the picture)
  3. ed by a distance from a reference point and an angle from a reference direction. When we think about plotting points in the plane, we usually think of rectangular coordinates (x,y) ( x, y) in the Cartesian coordinate plane

Introduction to polar coordinates. Double integrals (articles) Double integrals. Double integrals over non-rectangular regions. Double integrals beyond volume. Polar coordinates. This is the currently selected item. Double integrals in polar coordinates. Next lesson In order to work with complex numbers without drawing vectors, we first need some kind of standard mathematical notation.There are two basic forms of complex number notation: polar and rectangular. Polar Form of a Complex Number. The polar form is where a complex number is denoted by the length (otherwise known as the magnitude, absolute value, or modulus) and the angle of its vector (usually. 在opencv中,函数cvLogPolar功能是将图像映射到极坐标。 src 源图像dst 目标图像center 变换中心,此处输出精度最高。 M 幅度尺度参数flags:为插值方法标示与下面选项的组合:CV_WARP_FILL_OUTLIERS 填充目标图像中的所有像素,如果某些像素对应于源图像之我的位置,则用0填充.CV_WARP_INVERSE_MAP 表示矩阵是从. Fast Optic Disc Segmentation in Retina Using Polar Transform. Abstract: Glaucoma is one among major causes of blindness in working population. Early detection of Glaucoma through automated retinal image analysis helps in preventing vision loss. Optic disk (OD) segmentation from retinal images is the preliminary step in developing the diagnostic.

Transforms Overview. 09/04/2020; 7 minutes to read; a; In this article. This topic describes how to use the 2D Transform classes to rotate, scale, move (translate), and skew FrameworkElement objects.. What Is a Transform? A Transform defines how to map, or transform, points from one coordinate space to another coordinate space. This mapping is described by a transformation Matrix, which is a. An example using the cv::linearPolar and cv::logPolar operations. #include opencv2/imgproc.hpp. #include opencv2/highgui.hpp. #include <iostream>. using namespace cv; static void help ( void ) {. printf ( \nThis program illustrates Linear-Polar and Log-Polar image transforms\n. Usage :\n 22 votes, 14 comments. I spent around an hour playing around with interpolating between Cartesian and Polar forms of functions, I think I came up Visualisation of Cartesian - Polar Transformations

Comprehensive polar metabolomics and lipidomics profiling discriminates the transformed from the non-transformed state in colon tissue and cell line ever uses Fourier sine and cosine transforms. We practically always talk about the complex Fourier transform. Rather than separating f˜(k) into real and imaginary parts, which amounts to Cartesian coordinates, it is often helpful to write it as a magnitude and phase, as in polar coordinates. So we write f˜(k)=A(k)eiφ(k) (19) with A(k)= f˜(k

python - Converting an image from Cartesian to Polar

I am trying to transform an image (represented as a matrix) in R, into polar coordinate space with the origin being 0,0 (top left corner). Given the 215x215 matrix x which looks like:. x0 = as.vector(col(x)) y0 = as.vector(row(x)) r = sqrt( (x0^2) + (y0^2) )#x a = atan(y0/x0)#y m = as.matrix(data.frame(y=a, x=r)) m = round(m) m[m>215] = NA m[m==0] = NA xn = x[m] xn = matrix(xn, 215, 215 WGS84 NSIDC Arctic Polar Stereographic North (EPSG:3413) 0: 0: Location. R280 Learning & Environmental Sciences 1954 Buford Avenue Saint Paul, MN 55108 Find Us. Contact 612-626-050

LOG-POLAR TRANSFORM - 21084 - Robotpark ACADEMY

Polar Transform - roborealm

Coordinate Transformations Introduction We want to carry out our engineering analyses in alternative coordinate systems. Most students have dealt with polar and spherical coordinate systems. In these notes, we want to extend thi Many translated example sentences containing polar transform - English-German dictionary and search engine for English translations There have been a number of methods developed to sample from the Normal distribution including Inverse Transform Sampling, the Ziggurat Algorithm, and the Ratio Method (a rejection sampling technique). In this post we will focus on an elegant method called the Box-Muller transform. A quick review of Cartesian and polar coordinates Many translated example sentences containing polar transform - German-English dictionary and search engine for German translations In OpenCV, line detection using Hough Transform is implemented in the function HoughLines and HoughLinesP [Probabilistic Hough Transform]. This function takes the following arguments: edges: Output of the edge detector. lines: A vector to store the coordinates of the start and end of the line. rho: The resolution parameter in pixels

Using Polar and Log-Polar Transformations for Registration

Use Transform.Rotate to rotate GameObjects in a variety of ways. The rotation is often provided as an Euler angle and not a Quaternion. You can specify a rotation in world axes or local axes. World axis rotation uses the coordinate system of the Scene, so when you start rotate a GameObject, its x, y, and z axes are aligned with the x, y, and z. Polar-Fourier-Transform Description: A fast high accuracy Polar FFT algorithm is given in the software package written in MATLAB. For a given 2D signal of size n×n, the algorihtm s complexity is just like in a Cartesian 2D-FFT, which can be used widely in image processing and analysis

Resultant image from Figure 2 after the log-polar

In an improved Cartesian to polar coordinate transformation, a Cartesian system of discrete data elements is converted to a polar system of discrete data elements. Each Cartesian element is divided into subelements and one of the subelements is designated as the polar system origin. A polar system comprising an intersecting plurality of radial sector lines and a plurality of confocal arcs. Those results transform from components of polar to rectangular coordinates. If you want to go from rectangular to polar, you need r and θ in terms of x and y. $\endgroup$ - R.W. Bird. Oct 20 '20 at 14:15. Add a comment | 1 Answer Active Oldest Votes. 2. Transform Cartesian coordinates to polar or cylindrical. collapse all in page. Syntax [theta,rho] = cart2pol(x,y) [theta,rho,z] = cart2pol(x,y,z) Description. example [theta,rho] = cart2pol(x,y) transforms corresponding elements of the two-dimensional Cartesian coordinate arrays x and y into polar coordinates theta and rho Convert the polar equation into rectangular form: Possible Answers: Correct answer: Explanation: Recall that. Plugging this into the equation gives us. Multiply both sides by to get rid of the fraction. Recall that. So then the rectangular form of the equation is

Polar Transforms demo. An example using the cv.linearPolar and cv.logPolar operations.. This program illustrates Linear-Polar and Log-Polar image transforms. Sources List of books on the topic 'Quasi-polar transform'. Scholarly publications with full text pdf download. Related research topic ideas

Log-polar transformPolar Transform

Cognitive Polar Transformation. June 1, 2020 ·. Deep pain and hatred is being ignited. As the emotions rise to the surface, focus on those feelings. They are a powerful resource within yourself. Now.. draw that emotional energy deep into the center of your awareness so it can be transformed This basic capability is needed in various image analysis applications which include remote sensing, medical image analysis, object recognition, etc. In this paper, we have proposed an algorithm that is based on Log polar transform and phase correlation to register images which are transformed by rotation, translation and higher value of scale and the following images are the log-polar transformed magnitude plots of the two images: By performing phase correlation on the above two images, we get the following, with a peak at (11, -53): The log-polar images are translated by 11 and -53 pixels in x and y respectively Introduction. polarTransform is a Python package for converting images between the polar and Cartesian domain. It contains many features such as specifying the start/stop radius and angle, interpolation order (bicubic, linear, nearest, etc), and much more We have analysed 131 fragment-to-lead (F2L) examples targeting a wide variety of protein families published by academic and industrial laboratories between 2015-2019. Our assessment of X-ray structural data identifies the most common polar functional groups involved in fragment-protein binding are: N-H (hydr 2021 Chemical Science HOT Article Collectio

16.2.1 Transformations with coord_trans(). Like limits, we can also transform the data in two places: at the scale level or at the coordinate system level. coord_trans() has arguments x and y which should be strings naming the transformer or transformer objects (see Section 10).Transforming at the scale level occurs before statistics are computed and does not change the shape of the geom. Get the free Convert Complex Numbers to Polar Form widget for your website, blog, Wordpress, Blogger, or iGoogle. Find more Mathematics widgets in Wolfram|Alpha Roche's Maxim of Data Transformation. According to the internet, a maxim is a succinct formulation of a fundamental principle, general truth, or rule of conduct. [1] Maxims tend to relate to common situations and topics that are understandable by a broad range of people. Topics like data transformation Gradients in Polar Coordinates. If we apply formula (5) to polar coordinates with u1 = u r and u2 = u θ and use the derivative formulas in (4), we get ∇f =(Dur f)u r +(Du θ f)u θ = ∂f ∂r u r + 1 r ∂f ∂θ u θ = ∂f v r + 1 2 ∂f v θ. (6) This is much easier than the proof the author of our text has in mind for this formula in.