ME 2304: 3D Geometry & Vector Calculus Dr. Faraz Junejo Gradient of a Scalar field & Directional Derivative

Partial Derivatives  Let f(x,y) be a function with two variables.  If we keep y constant and differentiate f (assuming f is differentiable) with respect to the variable x, we obtain what is called the partial derivative of f with respect to x which is denoted by: Similarly If we keep x constant and differentiate f (assuming f is differentiable) with respect to the variable y, we obtain what is called the partial derivative of f with respect to y which is denoted by

Ex 1. Ex 2. Partial Derivatives: Examples

Ex 3. Partial Derivatives: Examples

Example 1: Find the partial derivatives f x and f y if f(x, y) is given by f(x, y) = x 2 y + 2x + y

Example 2: Find f x and f y if f(x, y) is given by f(x, y) = sin(x y) + cos x

Example 3: Find f x and f y if f(x, y) is given by f(x, y) = x e x y

Example 4: Find f x and f y if f(x, y) is given by f(x, y) = ln ( x y)

If f(x, y) = x 3 + x 2 y 3 – 2y 2 find f x (2, 1) and f y (2, 1) Example 5

Holding y constant and differentiating with respect to x, we get: f x (x, y) = 3x 2 + 2xy 3 – Thus, f x (2, 1) = = 16 Example 5 (contd.)

Holding x constant and differentiating with respect to y, we get: f y (x, y) = 3x 2 y 2 – 4y – Thus, f y (2, 1) = – 4. 1 = 8 Example 5 (contd.)

If calculate Exercise: 1

Using the Chain Rule for functions of one variable, we have: Exercise: 1(contd.)

Find f x, f y, and f z if f(x, y, z) = e xy l n z – Holding y and z constant and differentiating with respect to x, we have: – f x = ye xy l n z – Similarly, – f y = xe xy l n z – f z = e xy /z Exercise: 2

If f is a function of two variables, then its partial derivatives f x and f y are also functions of two variables. HIGHER DERIVATIVES

So, we can consider their partial derivatives (f x ) x, (f x ) y, (f y ) x, (f y ) y These are called the second partial derivatives of f. SECOND PARTIAL DERIVATIVES

If z = f(x, y), we use the following notation: NOTATION

Thus, the notation f xy (or ∂ 2 f/∂y∂x) means that we first differentiate with respect to x and then with respect to y. In computing f yx, the order is reversed. SECOND PARTIAL DERIVATIVES

Find the second partial derivatives of f(x, y) = x 3 + x 2 y 3 – 2y 2 – We know that f x (x, y) = 3x 2 + 2xy 3 f y (x, y) = 3x 2 y 2 – 4y Example 6

– Hence, Example: 6 (contd.)

Ex 3. Second-Order Partial Derivatives (f xx, f yy )

Exercise: 3

Notation for Partial Derivatives

Partial derivatives of order 3 or higher can also be defined. HIGHER DERIVATIVES

Calculate f xxyz if f(x, y, z) = sin(3x + yz) – f x = 3 cos(3x + yz) – f xx = –9 sin(3x + yz) – f xxy = –9z cos(3x + yz) – f xxyz = –9 cos(3x + yz) + 9yz sin(3x + yz) Example 7

Example: 8

Example: 8 (contd.)

Interpretations of Partial Derivatives As with functions of single variables partial derivatives represent the rates of change of the functions as the variables change. As we saw in the previous section, f x (x, y) represents the rate of change of the function f ( x, y) as we change x and hold y fixed while, f y (x, y) represents the rate of change of f ( x, y) as we change y and hold x fixed.

Scalar Field Every point in a region of space is assigned a scalar value obtained from a scalar function f(x, y, z), then a scalar field f(x, y, z) is defined in the region, such as the pressure or temperature in atmosphere, etc.

Examples of scalar quantities Altitude: Temperature: Electric potential: Pressure:

Scalar Field Scalar Field : A scalar quantity, smoothly assigned to each point of a certain region of space is called a scalar field Examples : i)Temperature and pressure distribution in the atmosphere ii) Gravitational potential around the earth

iii) Assignment to each point, its distance from a fixed point O Scalar Field (contd.)

O Once a coordinate system is set up, a scalar field is mathematically represented by a function : is the value of the scalar assigned to the point (x,y,z)

A smooth scalar field implies that the function,,is a smooth or differentiable function of its arguments, x,y,z. Scalar Field (contd.)

Since the scalar field has a definite value at each point, we must have O Consider two coordinate systems. O’

Gradient The gradient of a function, f(x, y), in two dimensions is defined as: The gradient of a function is a vector field. It is obtained by applying the vector operator to the scalar function f(x, y) Such a vector field is called a gradient (or conservative) vector field.

Gradient (contd.) Del operator Gradient Gradient characterizes maximum increase. If at a point P the gradient of f is not the zero vector, it represents the direction of maximum space rate of increase in f at P.

Example: 1 For the scalar field ∅ (x,y) = x 2 sin5y, calculate gradient of ∅

For the scalar field ∅ (x,y) = 3x + 5y, calculate gradient of f. Solution: Given scalar field ∅ (x,y) = 3x + 5y For the scalar field ∅ (x,y) = x 4 yz,calculate gradient of ∅. Example: 2

Gradient of a Scalar field vector field In vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change.

Interpretation Consider a room in which the temperature is given by a scalar field, T, so at each point (x,y,z) the temperature is T(x,y,z). At each point in the room, the gradient of T at that point will show the direction the temperature rises most quickly. The magnitude of the gradient will determine how fast the temperature rises in that direction.

Since temperature T depends on those three variables we can ask the question: how does T change when we change one or more of those variables? And as always, the answer is found by differentiating the function. In this case, because the function depends on more than one variable, we're talking partial differentiation. Gradient of temperature field

Now if we differentiate T with respect to x, that tells us the change of T in the x-direction. That is therefore the i-component of the gradient of T. You can see that there is going to be three components of the gradient of T, in the i, j and k directions, which we find by differentiating with respect to x, y and z respectively. So this quantity "the gradient of T" must be a vector quantity. Indeed it is a vector field. Gradient of temperature field (contd.)

This vector field is called "grad T" and written like and it is given as: Gradient of temperature field (contd.)

Gradient of temperature field : Summary In three dimensions, a scalar field is simply a field that takes on a single scalar value at each point in space. For example, the temperature of all points in a room at a particular time t is a scalar field. The gradient of this field would then be a vector that pointed in the direction of greatest temperature increase. Its magnitude represents the magnitude of that increase.

Example: 3 If T(x,y,z) is given by: Determine

Example: 4 Given potential function V = x 2 y + xy 2 + xz 2, (a) find the gradient of V, and (b) evaluate it at (1, -1, 3). Solution: (a) (b) Direction of maximum increase

Summary The gradient of a scalar field is a vector field, whose: Magnitude is the rate of change, and which points in the direction of the greatest rate of increase of the scalar field. If the vector is resolved, its components represent the rate of change of the scalar field with respect to each directional component.

Hence for a two-dimensional scalar field ∅ (x,y). And for a three-dimensional scalar field ∅ (x, y, z) Note that the gradient of a scalar field is the derivative of f in each direction Summary (contd.)

The gradient of any scalar field is a vector, whose direction is the direction in which the scalar increases most rapidly, and whose magnitude is the maximum rate of change Summary (contd.)

Directional Derivative

Directional Derivative: Example

Maximum and minimum value of Directional Derivative f Since, the directional derivative of f in the direction of n is just the scalar projection of grad f along the direction of n i.e.

Maximum and minimum value of Directional Derivative In other words, The maximum value of directional derivative is and it occurs when has the same direction as The minimum value of directional derivative is a and it occurs when has the opposite direction i.e.

Maximum and minimum value of Directional Derivative In other words, The maximum value of directional derivative is and it occurs when has the same direction as

Exercise: 1 Calculate the directional derivative of the following function in the given direction and at the stated point.

Exercise: 2 Calculate the directional derivative of the following function in the given direction and at the stated point.

Exercise: 2 (contd.)

Summary The directional derivative in any direction is given by the dot product of a unit vector in that direction with the gradient vector. So in effect, a directional derivative tells the slope of a surface in a given direction. f The directional derivative of f in the direction of n is just the scalar projection of grad f along the direction of n.