Partial Derivatives, Gradient, Divergence, and Curl

Partial Derivatives Recall that the derivative of a function of one variable provides information about how the function changes as the input variable changes. For a function of multiple variables, we consider how the function changes when one input variable changes and all other input variables are fixed, and refer to partial derivatives. For example, if f is a function of two variables x and y , then its partial derivatives are the functions fx and f y , which are defined by fxhy( +, ) - fxy( , ) fxyh( , +) - fxy( , ) fxyxy( , ) == lim and fxy( , ) lim hh ®®0hh 0 provided that the limit(s) exist.

Notation for Partial Derivatives For a function zfxy= ( , ), all of the following notation is used to represent partial derivatives. ¶¶fz ¶ fxyf( , ) == = fxy( , ) = == f DfDf = xx¶¶xx ¶ x11 x ¶¶fz ¶ fxyf( , ) == = fxy( , ) = == f DfDf = yy¶¶yy ¶ y22 y

Rule for Finding Partial Derivatives of zfxy= ( , )

1. To find fx , regard y as a constant and differentiate fxy( , ) with respect to x .

2. To find f y , regard x as a constant and differentiate fxy( , ) with respect to y .

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Partial derivatives are similarly defined for functions of three or more variables.

Higher Order Derivatives For higher order derivatives, we take derivatives one order at a time. For a function zfxy= ( , ), we use the notation

¶¶æöffz ¶22 ¶

()fffxx=== xx 11 ç÷ =22 = ¶¶xxèø ¶ x ¶ x

¶¶æöffz ¶22 ¶ ()fffxy=== xy 12 ç÷ = = ¶¶yxèø ¶¶¶¶ yxyx

¶¶æöffz ¶22 ¶ ()fffyx=== yx 21 ç÷ = = ¶¶xyèø ¶¶¶¶ xyxy

¶¶æöffz ¶22 ¶ ()fffyy=== yy 22 ç÷ =22 = ¶¶yyèø ¶ y ¶ y The notation is similar for higher order partial derivatives.

Clairaut’s Theorem Suppose that f is defined on a disk D that contains the point (ab, ). If the functions f xy and f yx are both continuous on D , then fabfabxy( , ) = yx ( , ). In essence, if f is a “nice” function, then a mixed partial derivative is independent of the order we take the derivatives, i.e. ffxy= yx .

Del Operator The del operator is denoted as Ñ which is called nabla. The del operator is a vector written as

æö¶¶¶æö æö Ñ =ç÷ijk +ç÷ + ç÷ èø¶¶¶xyzèø èø Basically, the presence of the Ñ tells us we are taking a derivate, which takes several forms, each of which gives different results! - If f is a function, then Ñf (gradient) is a vector.

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- If F is a vector field, then Ñ×F (divergence) is a scalar function. - If F is a vector field, then Ñ´F (curl) is a vector function. We will consider each of these forms below.

Gradient If f is a function of two variables x and y , then the gradient of f is the vector Ñf defined by ¶¶ff gradfxy( ,) = Ñ fxy( ,) = f( xyf ,) , ( xy ,) = i + j xy ¶¶xy where i and j are unit vectors.

The gradient of a function is similarly defined for functions of three or more variables. We think of the gradient of a function of two or more variables as the direction of the fastest increase of f at a point, with magnitude equal to the maximum rate of increase at a point – just as we do for the derivative of a single-variable function. Geometrically, the gradient at a point is perpendicular to the tangent vector of any curve that passes through the point. Example. If fxy( ,2) = xy23- 45 y+, find the gradient at the point (2, - 1).

Solution. Taking the required partial derivatives, we obtain

322 Ñfxy( ,,,) = fxy( xy) f( xy,4,) = xyxy 64.-

Plugging in the point (2, - 1), we obtain

Ñ-f (2, 1) = 4( 2)(- 1)322 ,6( 2) ( -- 1) 4=- 8, 20 .

Divergence ¶P If FPiQjRk=++, where P , Q , and R are functions, is a vector field on R3, and , ¶x ¶Q ¶R , and exist, then the divergence of F is defined by ¶y ¶z

¶¶¶PQR Ñ×FF =div = + + ¶¶¶xyz

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The divergence of a function is similarly defined for functions of an arbitrary number of variables. In essence, the divergence is the dot product of the del operator Ñ and the vector field F which results in a scalar function.

Example. If Fxzixyzj=+ , then find Ñ×F .

Solution. We note that R = 0 . Applying the definition, we obtain ¶¶¶PQR ¶ ¶ ¶ Ñ×Fxzxyzzxz = + + =( ) +( ) +(0.) = + ¶¶¶¶xyzx ¶ y ¶ z

Curl ¶P If FPiQjRk=++, where P , Q , and R are functions, is a vector field on R3, and , ¶x ¶Q ¶R , and exist, then the curl of F is defined by ¶y ¶z

æö¶¶RQæö ¶¶ PR æö ¶¶ QP Ñ´FF ==curl ç÷- i+ç÷- j+ ç÷- k èø¶¶yzèø ¶¶ zx èø ¶¶ xy

ijk ¶¶¶ Or equivalently, Ñ´F = . ¶¶¶xyz PQR

The curl of a function is not defined for functions with a different number of variables; it only applies to 3-dimensional space.

In essence, the curl is the cross product of the del operator Ñ and the vector field F which results in a vector field.

Geometrically, the curl of F is a vector that is perpendicular to the vector field that represents the velocity of a particle in motion. Its length and direction characterize the rotation at a particular point.

Example. If Fxyzixyjxyzk=22-+ 23, then find Ñ´F .

Solution. Applying the definition, we obtain

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ijk ¶¶¶ Ñ´F = ¶¶¶xyz PQR ijk ¶¶ ¶ = ¶¶xy ¶ z xyz2223- x y x yz

æö¶¶23 2æö ¶¶ 2 23 æö ¶ 2 ¶ 2 =ç÷xyz--( xy) i+ç÷ xyz- xyz j+ ç÷()-- xy xyz k èø¶¶yzèø ¶¶ zx èø ¶ x ¶ y =+( xz23) i(22 xyz- xyz 3) j+(-- 2 xy xz 2) k .

Fun Facts: Theorem. If f is a function of three variables that has continuous second-order partial derivatives, then Ñ´Ñff =curl (Ñ) = 0.

It follows that if F is conservative, meaning there is a function f such that Ff=Ñ , then Ñ´FF =curl( ) = 0.

Theorem. If FPiQjRk=++ is a vector field on R3 that has continuous second-order partial derivatives, then Ñ×( Ñ´FF) = d i v c u r l = 0

Proof.

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ijk ¶¶¶ Ñ×( Ñ´F ) =Ñ× ¶¶¶xyz PQR æöæö¶¶RQæö ¶¶ PR æö ¶¶ QP =Ñ×ç÷ç÷ -ijk +ç÷-+ ç÷- èøèø¶¶yzèø ¶¶ zx èø ¶¶ xy ¶¶æöRQ ¶ ¶¶æö PR ¶ ¶¶ æö QP ¶ =ç÷-+ç÷-+ ç÷- ¶¶xyèø ¶ z ¶¶ yzèø ¶ x ¶¶ zx èø ¶ y ¶¶¶¶¶¶222222RQPRQP = -+-+-= 0, ¶¶xy ¶¶ xz ¶¶ yz ¶¶ yz ¶¶ zx ¶¶ zx since mixed partial derivatives do not depend on the order of the variables.

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