Math 241: Multivariable calculus Professor Leininger Fall 2014
Calculus of 1 variable In Calculus I and II you study real valued functions of a single real variable. Examples: f (x) = x 2, r(x) = 2x2 +x g(u) = e u,... x 3 5x+20 y = f (x), h(θ) = sin(θ) + cos(2θ), T (t) = temperature in Champaign-Urbana, t hours after midnight on August 25. ρ(d) = density of a piece of wire at distance d from one end.
Three key concepts from Calculus I, II. f (x), a function of one variable. 1. The derivative: f (x) = df dx = d dy dx f (x) = dx. Rate of change. Slope of the tangent line to the graph. 2. The integral: b a f (x) dx. Signed area under graph. 1 b Average value b a f (x) dx. a 3. Fundamental Theorem of Calculus: Relates the two. f (b) f (a) = b a f (x) dx.
1 variable is too constrained Functions of a single variable are insufficient for modeling more complicated situations. Examples: The temperature depends on location as well as time. Need to specify location, e.g. by latitude x and longitude y, and time, e.g. t hours after midnight: T (x, y, t) = temperature at time t in location (x, y). Density of a flat sheet of metal can depends on the point in the sheet, specified by x and y coordinates δ(x, y) = density of point at (x, y) in a sheet of metal
The setting: n dimensional space, R n. R 1 = 1 dimensional space = R = real line R 2 = 2 dimensional space = Cartesian plane = {(x, y) x, y R} = ordered pairs of real numbers. R 3 = 3 dimensional space = {(x, y, z) x, y, z R} = ordered triples of real numbers. The numbers x, y in R 2 or x, y, z in R 3 are the coordinates of the point.
The setting: n dimensional space, R n. For any n = 1, 2, 3, 4,..., we have R n = n dimensional space = (x 1, x 2,..., x n ) x i R} = ordered n tuples real numbers. x 1,..., x n are the coordinates of the point. We will study functions whose domain (and range) is a subset of R n. Dimensions 1,2 and 3 will serve as motivation and provide intuition, though much of the theory works for all n (but not all!)
Plan for this course Develop calculus to study functions of several variables. 1. Derivatives: Chapter 14 (and 13) 2. Integrals: Chapter 15 (and 13) 3. Fundamental Theorems of Calculus : Chapter 16
What do we need in order to do calculus? Question: What sets calculus apart from algebra, trigonometry, pre-calculus? Answer: Limits! In one variable: lim f (x) = L means x a as x approaches a, f (x) approaches L This requires a notion of proximity and hence of distance.
Distance Given P(x 1,..., x n ), Q(y 1,..., y n ) R n, define the distance between these points to be PQ = distance from P to Q = (x 1 y 1 ) 2 + (x 2 y 2 ) 2 +... + (x n y n ) 2 n = 2 (Pythagorean Theorem) Distance from (a, b) to (c, d) is (a c) 2 + (b d) 2 z (p,q,r) n = 3 Distance from (a, b, c) to (p, q, r) is (a p) 2 + (b q) 2 + (c r) 2 x (a,b,c) y
Spheres: application of distance formula For C(a, b, c) R 3 and r > 0, the sphere of radius r with center C is the set Then {P R 3 PC = r} PC = r (x a) 2 + (y b) 2 + (z c) 2 = r (x a) 2 + (y b) 2 + (z c) 2 = r 2 This is equation for sphere: set of points (x, y, z) satisfying this equation is a sphere. Compare equation of a circle (x a) 2 + (y b) 2 = r 2.
What else do we need to do calculus? Derivatives (and integrals) require limit and displacement: f f (x + h) f (x) (x) = lim. h 0 h Need displacement from x to x + h (and from f (x) to f (x + h)). This requires vectors...
Vectors in R 2. A vector in R 2 is an arrow. It represents a quantity with both direction and magnitude. Two vectors are equal if they have the same direction and magnitude. Notation: book v (bold face) or written v (arrow over).
Operations and constructions Addition: Parallelogram rule. Scalar multiplication: Scale magnitude, and reverse direction if negative. Displacement vector from A to B is AB, tail at A and tip at B.
Position vectors Every vector is equal to one with tail at the origin O = (0, 0): Vectors in R 2 R 2 OP P OP = position vector of the point P. (vectors and points are different objects) Write v = v 1, v 2 (or sometimes just (v 1, v 2 )). The numbers v 1, v 2 are the components of v.
Calculations in terms of components For u = (u 1, u 2 ), v = (v 1, v 2 ), and c R we have u + v = (u 1 + v 1, u 2 + v 2 ) c u = (cu 1, cu 2 ) u = u1 2 + u2 2 = magnitude of u. Sometimes write u instead. Also call it the norm or length of u. A(a 1, a 2 ) and B(b 1, b 2 ), the displacement vector is AB = (b 1 a 1, b 2 a 2 ).
Vectors in R n Vectors in R n R n OP P OP = position vector of the point P. v = v 1,..., v n = position vector of the point (v 1,..., v n ). v 1,..., v n are the components. Displacement vector from point A(a 1,..., a n ) to B(b 1,..., b n ): Arrow from A to B AB = (b 1 a 1,..., b n a n ).
Physical quantities We will also use forces to denote physical quantities: Example: Forces have a direction and magnitude, so we can represent them with vectors Multiple forces acting on an object, then the net force on object (or resultant force) is the sum of the forces. Example: Wind has speed and direction, so can represent it with a vector.