Local Linearity (Tangent Plane) Unit #19 : Functions of Many Variables, and Vectors in R 2 and R 3

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1 Local Linearity and the Tangent Plane - 1 Unit #19 : Functions of Many Variables, and Vectors in R 2 and R 3 Goals: To introduce tangent planes for functions of two variables. To consider functions of more than two variables and their level surfaces. To study the differential of a function and its interpretation as the linear approximation of measurement error. To learn about vectors. Local Linearity (Tangent Plane) Reading: Section Just as a graph of a function of a single variable looks like a straight line when you zoom in to a point, the graph of a function f of two variables looks flat when you zoom in to a point (a, b, f(a, b)) on it. Reading: Sections 12.5, 14.3, 13.1, and Let us write the equation of the tangent plane to z = f(x, y) in point/slope form: z = c + m(x a) + n(y b), What is the slope of f(x, y) in the x direction at (a, b)? Local Linearity and the Tangent Plane - 2 Local Linearity and the Tangent Plane - 3 Based on those calculations, if we are given f(x, y), and want the tangent plane at (a, b), what values should we pick for the slopes m and n in the tangent plane formula? What is the slope of the tangent plane in the x direction at (a, b)? If the tangent plane passes through (a, b, f(a, b)), what value do we need for c in z = c + m(x a) + n(y b)?

2 Local Linearity and the Tangent Plane - 4 Local Linearity and the Tangent Plane - 5 Tangent Plane The plane tangent to z = f(x, y) at the point (x, y) = (a, b) is defined by z = f(a, b) + f x (a, b)(x a) + f y (a, b)(y b) This can be used to define the local linear approximation to f(x, y) at points (x, y) near (a, b): An alternate form of this same relationship is f f x (a, b) x + f y (a, b) y We will revisit use these two forms interchangeably, depending on whether we want to calculate local changes in f(x, y), or the estimated values of f(x, y). f(x, y) f(a, b) + f x (a, b)(x a) + f y (a, b)(y b) Note the similarity of the right hand side of a tangent line formula for single variable functions. Tangent Plane Examples - 1 Tangent Plane Examples Example: What is the equation of the tangent plane to the graph of the function f(x, y) = 3xe y/2 at (1, 2, 3e)? Tangent Plane Examples - 2 Use your answer to the preceding question to find an approximate value for f(0.95, 2.03) and use a calculator to check the accuracy of your answer.

3 Tangent Plane Examples Example: Let f(x, y) = 2 + x 2 2x + y2. Find the points on the graph where the tangent plane is parallel to the (x, y)-plane (i.e. horizontal). Sketch what the surface might look like near this/these points. Tangent Plane Examples - 4 [See also examples 3, 4, 5 in Section 14.3.] Functions of More Than Two Variables - 1 Functions of More Than Two Variables - 2 Functions of More Than Two Variables Reading: Section Example: (Temperature in a Room) To specify a point in the room, you must specify three coordinates (x, y, z). So, the temperature is a function of three variables: T = T (x, y, z). What is another variable that we could potentially include in this function, that would make temperature a function of four variables? Pictures for a Function of Three Variables To picture f(x, y, z) by means of a graph is not very useful. The graph would have to have the following definition: {(x, y, z, w) : w = f(x, y, z)}. That is, it would be a set in 4 dimensional space, which is not something we can directly visualize. However, we can make use of the analogue of the contour diagram. Suppose we pick a constant C and collect all the points (x, y, z) for which f(x, y, z) = C. In general, this gives a surface called a level surface. By sketching a number of level surfaces, we can get a sense of the function f. [See also H-H, Examples 2-5 in Section 12.5.]

4 Functions of More Than Two Variables - 3 Functions of More Than Two Variables - 4 Example (Steam Heating) Queen s University is heated by steam which is distributed to the buildings via underground pipes. Depict the level surfaces for the temperature function near the steam pipe under Agnes Benidickson Field. Contour of fixed density in a CT scan A Simple and Flexible Volume Rendering Framework for Graphics-Hardware-based Raycasting, Stegmeier et. al. Material contours based on MRI of a head. isocaps from Mathworks/MATLAB. Schlumberger Water Services - Groundwater Modelling Software MT3D99 A program for generating electron density isosurfaces for presentation in protein crystallography. M. C. Lawrence 1, P. D. Bourke Partial Derivatives in Higher Dimensions - 1 Partial Derivatives in Higher Dimensions If g(x, y, z) = x2y + 3yezy g, find. 2x y In general, for a function of several variables f (x1,..., xn), the partial derivative f (x1,..., xn) xi is the derivative obtained by holding all other variables fixed. f. z GMU s EastFire Cluster - Real-time wind velocities for fire spread prediction. Xianjun Hao and John J. Qu Partial Derivatives in Higher Dimensions - 2 General Partial Derivatives If f (x, y, z) = xeyz + z 2 cos(x2y), find Visualization of Fluid Turbulence using AVS/Express, CEI/Ensight and Paraview. P.K. Yeung et. al. If h(x, y, z) = sin(xy) + xz 3, find h (0, 2, 3). x

5 Local Linearity for a Function of Three Variables Local Linearity for a Function of Three Variables - 1 The local linearity of f(x, y, z) for points (x, y, z) near (a, b, c) is described by an analogous formula to the one used for functions of two variables: Example: for Local Linearity for a Function of Three Variables - 2 Use local linearity around (6, 3, 1) to find an approximate value (6.02)2 + (2.96) 3 + (1.01) 4. f(x, y, z) f(a, b, c) + f x (a, b, c)(x a) + f y (a, b, c)(y b) + f z (a, b, c)(z c) The Differential - 1 The Differential - 2 The Differential Reading: Section Two forms of the two-variable local linearity we have seen are f(x, y) f(a, b) + f x (a, b) (x a) + f y (a, b) (y b) and f f x (a, b) x + f y (a, b) y If x and y are allowed to get smaller and smaller, this approximate inequality becomes more nearly accurate. In the limit, mathematicians change the values into d values and call them differentials. In the limit, we obtain an expression with an equality instead of an approximation: df = f x (a, b) dx + f y (a, b) dy Conceptually, most people think of this as a formula for change in f for very small changes in x and y.

6 The Differential - 3 For a function of three variables, f(x, y, z), the local linearity can be expressed in similar ways: f(x, y, z) f(a, b, c) + f x (a, b, c) (x a) + f y (a, b, c) (y b) + f z (a, b, c) (z c) Using Linearity/Differentials for Estimates of Error Bounds - 1 Using Linearity/Differentials for Estimates of Error Bounds In scientific experiments, it is important to be able to provide estimates of the accuracy of your measurements and to derive estimates of the accuracy of your conclusions. Differentials, or linear approximations, are the key to making this kind of calculation simple. f f x (a, b, c) x + f y (a, b, c) y + f z (a, b, c) z df = f x (a, b, c) dx + f y (a, b, c) dy + f z (a, b, c) dz Error Bounds For a 2-variable function, local linearity gives us f f x (a, b) x + f y (a, b) y Using Linearity/Differentials for Estimates of Error Bounds - 2 To avoid sign confusion, errors are often given in the form (measured value) ± (error). This means that, if we measured x, x < (max x error). In these cases we can estimate the maximum (positive) error in the function output using error bound f x (a, b) (max x) + f y (a, b) (max y) The same result applies to 3- and higher-variable functions, with an additional term for each variable. Example: radius 2.5 cm height 15.0 cm mass 330 g Using Linearity/Differentials for Estimates of Error Bounds - 3 Suppose we measure the following properties of a cylinder: Suppose further that the accuracy of our length measurements is estimated to be ±0.1 cm and the accuracy of our mass measurement to be ±2 g. Calculate the density of the cylinder.

7 Using Linearity/Differentials for Estimates of Error Bounds - 4 Now estimate the error bound in this calculation. Include units. Using Linearity/Differentials for Estimates of Error Bounds - 5 If a is the error made in a measurement (or calculation) of a quantity a, then the ratio a is often referred to as the relative error. If we multiply it by 100, as a in a 100, we get the percentage error. a Suppose f(x, y, z) = x m y n z p. Show that the following formula for relative error is correct. f f m x x + n y y + p z z (CAUTION: This relative error formula only holds for functions that have this special form!) Vectors Vectors - 1 Reading: Sections We have seen that partial derivatives of a multi-variate function let us compute the slope of a surface in the x and y directions. However, we can also define a slope or steepness if we choose to move in an arbitrary direction. To do that kind of computation, we will need to introduce notation for arbitrary (x, y) directions. We will find the vector representation very helpful in this regard. Vectors - 2 It is often convenient to use arrows to represent vector quantities. The length of the arrow corresponds to the magnitude of the vector and the direction of the arrow tells you the direction of the vector. A natural question to ask is does it matter where the vector is located? The answer to this question is that it does not matter. Two arrows that have the same direction and magnitude are two representations of the same vector. Vectors Scalars A vector is a quantity that has both magnitude and direction. Velocity and force are examples of vector quantities. Quantities that have magnitude only are called scalars. Length and volume are examples of scalar quantities.

8 Vectors - 3 Which of the following arrows represent the same vector (i) the arrow from (0, 0) to (3, 4); (ii) the arrow from (2, 2) to (5, 2); (iii) the arrow from ( 1, 3) to (2, 7); (iv) the arrow from (1, 2) to ( 2, 2)? Vectors - 4 It is possible to combine vector quantities in much the same way as we combine numbers (by addition and subtraction). For the vectors v 1 and v 2 below v 1 v 2 sketch v 1 + v 2 v 1 v 2 v 2 v 1 Vectors - 5 Vector Components and Magnitudes - 1 Sketch the vectors ( v 1 v 1 ), and ( v 2 v 2 ). Vector Components and Magnitudes To manipulate vectors without needing to draw arrows, we need a symbolic representation for them. One of the most useful representations is the component form of a vector. The result of these last two differences is called the zero vector. It is a vector with zero length, and is the only vector for which we can assign no direction.

9 Component Unit Vectors We define i a vector of length 1 in the direction of the x axis j a vector of length 1 in the direction of the y axis k a vector of length 1 in the direction of the z axis Components of a vector If we expression a vector in the form v = v 1 i + v 2 j + v 3 k we call v 1 i, v 2 j, and v 3 k the components of v Alternate Component Form If v = v 1 i + v 2 j + v 3 k, a shorter form for the component representation is v = }{{} v 1, }{{} v 2, }{{} v 3 i j k Vector Components and Magnitudes - 2 Vector Components and Magnitudes - 3 Note: sometimes mathematicians and other scientists use different bracket shapes v = [v 1, v 2, v 3 ] or v = (v 1, v 2, v 3 ) to indicate that the set of values represents a vector, rather than a point. We will continue to use either the vector components, i, j and k, or angled parentheses, and. Vector Components and Magnitudes - 4 Vector Components and Magnitudes - 5 Example: Express the following vectors in both component forms: v 1 v 3 v 4 v 1 v 3 v 4 v 2 v 2 Question: Which of the following vectors represents v 1 + v 3? (a) 2, 3 (b) 3, 3 (c) 3, 2 (d) 3, 3 Question: Which of the following vectors represents v 4 v 3? (a) 3, 3 (b) 3, 5 (c) 1, 1 (d) 1, 1

10 Magnitude or Length From Components If v = v 1 i + v 2 j The length of a vector v = v1 2 + v2 2 In 3 dimensions, where v = v 1 i + v 2 j + v 3 k, v = v1 2 + v2 2 + v2 3 Vector Components and Magnitudes - 6 Parallel Vectors Parallel Vectors - 1 It is often important to know whether two vectors are parallel to each other, that is that they point along the same line, but might have different magnitudes. Example: Are the vectors v = 1, 2 and w = 2, 4 parallel? How long is the vector 2, 3? (a) -1 (b) 5 (c) 13 (d) 13 Why are the vectors v = 1, 2 and w = 2, 5 not parallel? Parallel Vectors - 2 Parallel Vectors Two vectors, v and u, are parallel to one another if there exists a scalar multiplier c such that v = c u. Parallel Vectors - 3 Example: Find the value(s) of t such that the vector u = 2, 3 t, 3 + t 2 is parallel to w = 4, t 2 9, 4t. Example: Show that the vector 3, 4, 5 is not parallel to 1, 2, 3.

11 Vector Applications Vector Applications - 1 Example: An airplane sets itself on a heading due east, at an air speed of 600 km/hr. It has a tail-wind of 80 km/h from the north-west. What is the plane s net velocity vector relative to the ground? Vector Applications - 2 Example: An airplane is flying at an airspeed of 800 km/hr, in winds that are blowing from the south at a speed of 50 km/hr. In what direction should the plane head to end up going due east, relative to the ground?