Lecture 9: functions of several variables Content: - basic definitions and properties - partial and total differentiation - differential operators - multiple integrals - examples of multiple integrals Functions of several variables: In more rigorous mathematical language: Functions of several variables: examples of graphs for f = f(x,y) another kind of visualization - so called coloured image maps (there exist also so called contour maps) another kind of visualization - so called coloured image maps (there exist also so called contour maps) functions f = f(x,y,z) are often visualized in form of voxel maps Functions of several variables: Functions of several variables are used in science for the description of various fields (physical fields, fields of properties ...). scalar fields: e.g. temperature, density, concentration, electric charge, ... t(x,y,z), ρ(x,y,z), U(x,y,z),... and also vector fields: e.g. electrical intensity, fluid velocity, gravitational acceleration,... A = [Ax ,Ay ,Az ] Ax = Ax (x, y,z) Ay = Ay (x, y,z) Az = Az (x, y,z) Functions of several variables: Many properties are identical with the case of a function with one variable. Functions of several variables: Many properties are identical with the case of a function with one variable. With the continuity is connected also the so called distance function d: Functions of several variables: Some properties are new (compared with a function with one variable). Symmetry: A symmetric function is a function f is unchanged when two variables xi and xj are interchanged: where i and j are each one of 1, 2, ..., n. For example: is symmetric in x, y, z since interchanging any pair of x, y, z leaves f unchanged, but is not symmetric in all of x, y, z, t, since interchanging t with x or y or z is a different function. Lecture 9: functions of several variables Content: - basic definitions and properties - partial and total differentiation - differential operators - multiple integrals - examples of multiple integrals Functions of several variables: Some properties are new (compared with a function with one variable). Partial derivatives: In the case of functions of several variables, we recognize: a) total derivative (all variables can vary and derivatives with respect to all variables are involved) b) partial derivative (it is a derivative with respect to one of the variables with the others held constant) In the notation, Leibniz rule is used more often (symbol ∂ is derived from “d” and it was introduced by Legendre – it is called as partial derivative symbol). Example, function f = x2+xy+y2: ∂f ∂ = (x2 + xy + y 2 )= 2x + y + 0 = 2x + y ∂x ∂x ∂f ∂ = (x2 + xy + y 2 )= 0 + x + 2y = x + 2y ∂y ∂y another tool is given in the next slide: Partial derivatives: For the beginner it is helpful to imagine instead of a variable (e.g. y) for a moment a constant (e.g. b). Partial derivatives – few examples: Functions of several variables: Some properties are new (compared with a function with one variable). Total derivative (differential): In the case of functions of several variables, we recognize: a) total derivative (all variables can vary and derivatives with respect to all variables are involved) b) partial derivative (it is a derivative with respect to one of the variables with the others held constant) Example, function f = x2+xy+y2: ∂ ∂ df = (x2 + xy + y 2 )dx + (x2 + xy + y 2 )dy = (2x + y)dx + (2y + x)dy ∂x ∂y Differential operators: There exist few special operations, which use partial derivatives and express properties of analyzed functions of several variables – so called differential operators: - gradient (grad) - divergence (div) - rotation (rot) - Laplacian operator (divgrad) These are used in various descriptions and derivations of basic properties of physical fields. Differential operators: Gradient – show the direction and size of the greatest change of a scalar field in each point of its domain, input of the operation: scalar field output of the operation: vector field ∂U ∂U ∂U gradU = i + j + k ∂x ∂y ∂z where i , j , k are elementary vectors (pointing in the direction of each coordinate axis – see 2. lecture, 26.slide) Comment to the notation: We can write gradient using the so called nabla or del operator ∇: ∂ ∂ ∂ gradU = ∇U where ∇ = i + j + k ∂x ∂y ∂z Differential operators: Gradient – show the direction and size of the greatest change of a scalar field in each point of its domain. ∂U ∂U ∂U gradU = i + j + k ∂x ∂y ∂z In physical fields, gradient is always pointing in the direction of force lines (perpendicular to equipotential lines). Gradient – example (field of positive electrical charge): (1/3) Electrical potential U, caused by a positive electrical point charge (Q), situated in the origin of the coordinate system (Cartesian) can be described by means of the following equation: 1 Q 1 Q U = = 2 2 2 4πε0 r 4πε0 x + y + z -12 where ε0 is the electrical permitivity of vacuum (8.854⋅10 F/m). Equipotential surfaces of this scalar field build spherical surfaces around the origin of the coordinate system. Gradient is a vector field, which vectors point in each point of the space perpendicular to these equipotential surfaces. Gradient – example (field of positive electrical charge): (2/3) ∂U ∂U ∂U gradU = i + j + k ∂x ∂y ∂z We will evaluate the gradient of this scalar function: 1 Q 1 Q U = = 2 2 2 4πε0 r 4πε0 x + y + z because the field of electrical intensity (vector) is given: E = −gradU First we evaluate the partial derivatives of U with respect to x, y and z. ∂U Q ∂ −1 2 Q 1 −3 2 = [x2 + y2 + z2 ] = − [x2 + y2 + z2 ] 2x = ∂x 4πε0 ∂x 4πε0 2 Q x Q x = − = − πε 2 2 2 3 2 πε 3 4 0 [x + y + z ] 4 0 r ∂U Partial derivatives and ∂ U are evaluated in a very similar way. ∂y ∂z Gradient – example (field of positive electrical charge): (3/3) ∂U Q x ∂U Q y ∂U Q z = − = − = − 3 , 3 , 3 ∂x 4πε0 r ∂y 4πε0 r ∂z 4πε0 r Q x y z Q xi + yj + zk Q r = − = + + = = E gradU 3 i 3 j 3 k 3 3 4πε0 r r r 4πε0 r 4πε0 r This is a vector field, pointing in the same direction as the vector r and having the size: Q r Q 1 = = E 3 2 4πε0 r 4πε0 r Differential operators: There exist few special operations, which use partial derivatives and express properties of analyzed functions of several variables – so called differential operators: - gradient (grad) - divergence (div) - rotation (rot) - Laplacian operator (divgrad) These are used in various descriptions and derivations of basic properties of physical fields. Differential operators: Divergence – tells about the sources of a vector field: when the result is zero then there is no source of the field in the point. input of the operation: components of vector field output of the operation: scalar value field ∂A ∂A ∂A divA = x + y + z ∂x ∂y ∂z where Ax, Ay, Az are the components of vector A . Comment: Divergence depends on the changes of the size of vector components and not the change of their direction. Comment to the notation: We can write also divergence using the nabla or del operator ∇: ∂ ∂ ∂ divA = ∇ ⋅ A where ∇ = i + j + k ∂x ∂y ∂z Divergence – example (field of electrical charge): (1/2) Field of electrical intensity (a vector field) is given by: Q xi + yj + zk = − = = + + E gradU 3 Exi Ey j Ezk 4πε0 r Q x Q y Q z = = = Ex 3 , Ey 3 , Ez 3 4πε0 r 4πε0 r 4πε0 r To evaluate the divergence of this field, we need to evaluate the following derivatives: ∂E Q ∂ x Q ∂ x Q ∂ −3 2 x = = = x[x2 + y2 + z2 ] = ∂ πε ∂ 3 πε ∂ 2 2 2 3 2 πε ∂ x 4 0 x r 4 0 x [x + y + z ] 4 0 x Q −3 2 − 3 −5 2 = [x2 + y2 + z2 ] + x [x2 + y2 + z2 ] 2x = 4πε0 2 Q −3 2 −5 2 = [x2 + y2 + z2 ] − 3x2 [x2 + y2 + z2 ] 4πε0 Divergence – example (field of electrical charge): (2/2) For all three derivatives we get: ∂E Q −3 2 −5 2 x = [x2 + y2 + z2 ] − 3x2 [x2 + y2 + z2 ] ∂x 4πε0 ∂E Q −3 2 −5 2 y = [x2 + y2 + z2 ] − 3y2 [x2 + y2 + z2 ] ∂y 4πε0 ∂E Q −3 2 −5 2 z = [x2 + y2 + z2 ] − 3z2 [x2 + y2 + z2 ] ∂z 4πε0 ∂E ∂E ∂E Q −3 2 −5 2 x + y + z = 3[x2 + y2 + z2 ] − 3(x2 + y2 + z2 )[x2 + y2 + z2 ] = ∂x ∂y ∂z 4πε0 Q −3 2 −3 2 = 3[x2 + y2 + z2 ] − 3[x2 + y2 + z2 ] = 0 4πε0 This result is valid for all points with the exception of the coordinate system origin, where x = y = z = 0 (source area). Differential operators: There exist few special operations, which use partial derivatives and express properties of analyzed functions of several variables – so called differential operators: - gradient (grad) - divergence (div) - rotation (rot) - Laplacian operator (divgrad) These are used in various descriptions and derivations of basic properties of physical fields. Differential operators: Rotation – tells about the existence of so called curls of the vector field (not about the sources). input of the operation: components of vector field output of the operation: vector field i j k ∂ ∂ ∂ ∂ ∂ ∂Az Ay ∂Ax ∂Az Ay ∂Ax rotA = = i − + j − + k − ∂x ∂y ∂z ∂y ∂z ∂z ∂x ∂x ∂y Ax Ay Az Comment: Rotation does not depend on the changes of the size of vector components (this was the role of divergence). Comment to the notation: We can write also divergence using the nabla or del operator ∇: ∂ ∂ ∂ rotA = ∇× A where ∇ = i + j + k ∂x ∂y ∂z Rotation – example (field of electrical charge): (1/2) Field of electrical intensity (a vector field) is given by: Q xi + yj + zk = − = = + + E gradU 3 Exi Ey j Ezk 4πε0 r Q x Q y Q z = = = Ex 3 , E y 3 , Ez 3 4πε0 r 4πε0 r 4πε0 r For the rotation evaluation we need following derivatives: ∂E Q ∂ z Q ∂ z zQ ∂ −3 2 z = = = [x2 + y2 + z2 ] = ∂ πε ∂ 3 πε ∂ 2 2 2 3 2 πε ∂ y 4 0 y r 4 0 y [x + y + z ] 4 0 y zQ − 3 −5 2 − 3yzQ −5 2 = [x2 + y2 + z2 ] 2y = [x2 + y2 + z2 ] 4πε0 2 4πε0 ∂E Q ∂ y − 3zyQ −5 2 y = = [x2 + y2 + z2 ] ∂ πε ∂ 2 2 2 3 2 πε z 4 0 z [x + y + z ] 4 0 Rotation – example (field of electrical charge): (2/2) From the evaluated derivatives it follows: ∂E ∂E z − y = 0 ∂y ∂z In a similar way we can show: ∂E ∂E ∂E ∂E x − z = 0 y − x = 0 ∂z ∂x ∂x ∂y ..