Vector Calculus One-Shot Revision Notes: Vector Differentiation

Instructor: Dr. Gajendra Purohit.
Purpose: To provide a comprehensive summary of four key videos on Vector Differentiation within the Vector Calculus domain, specifically designed for university students needing an exhaustive revision in approximately 10 to 12 minutes.
Topics Covered: * Basic concepts (Scalar and Vector Point Functions). * Gradient of a function. * Directional Derivatives. * Unit Normal Vectors. * Angles between vectors/surfaces. * Divergence and Curl.

Region Definition: Consider a point $(x, y, z)$ within a region $r$ .
Scalar Point Function: * If there is a unique scalar corresponding to every point in the region, the function is a scalar point function. * A scalar function does not involve direction; therefore, it does not contain the unit vectors $\mathbf{i}$ , $\mathbf{j}$ , or $\mathbf{k}$ . * Equation form: Typically represented as a simple algebraic equation in terms of $x$ , $y$ , and $z$ .
Vector Point Function: * Involves direction, meaning the function includes the unit vectors $\mathbf{i}$ , $\mathbf{j}$ , and $\mathbf{k}$ . * Example representation: $\mathbf{F} = x^2 y ext{ } \mathbf{i} + 2xy ext{ } \mathbf{j} + z^2 ext{ } \mathbf{k}$ .

Derivative of Scalar Product (Product Rule): * If a scalar is given in product form, the derivative is calculated using the $u \times v$ rule. * Formula: $\frac{d}{dt}(a − b) = a − \frac{db}{dt} + b − \frac{da}{dt}$ . * The result of a scalar product derivative is a scalar quantity.
Derivative of Vector Product (Cross Product): * Calculated similarly using the product rule but maintains vector directionality. * Formula: $\frac{d}{dt}(\mathbf{a} \times \mathbf{b}) = \mathbf{a} \times \frac{d\mathbf{b}}{dt} + \frac{d\mathbf{a}}{dt} \times \mathbf{b}$ . * The result of a vector product derivative is a vector quantity.

The Del Operator (\nabla): * Defined as: \nabla = \mathbf{i} \frac{\partial}{\partial x} + \mathbf{j} \frac{\partial}{\partial y} + \mathbf{k} \frac{\partial}{\partial z}.
Gradient Formula: * When applied to a scalar function $f$ , it results in: $\text{grad } f = \nabla f = \mathbf{i} \frac{\partial f}{\partial x} + \mathbf{j} \frac{\partial f}{\partial y} + \mathbf{k} \frac{\partial f}{\partial z}$ . * This represents a vector that can be used to calculate normal vectors, unit normal vectors, and directional derivatives.
Example Calculation: * Function: $f = e^{x^2 + y^2 + z^2}$ . * Step 1: Calculate partial derivatives. * $\frac{\partial f}{\partial x} = 2x e^{x^2 + y^2 + z^2}$ * $\frac{\partial f}{\partial y} = 2y e^{x^2 + y^2 + z^2}$ * $\frac{\partial f}{\partial z} = 2z e^{x^2 + y^2 + z^2}$ * Step 2: Combine into gradient form: $∇f = 2e^{x^2 + y^2 + z^2} (x\mathbf{i} + y\mathbf{j} + z\mathbf{k})$ . * Step 3: Evaluate at point $(1, 1, 1)$ : $∇f|_{(1,1,1)} = 2e^3 (\mathbf{i} + \mathbf{j} + \mathbf{k})$ .

Directional Derivative (D.D.): * The rate of change of a function in a specific direction defined by a vector $\mathbf{a}$ . * Formula: ∇f \cdot \mathbf{\hat{a}}, where $\mathbf{\hat{a}}$ is the unit vector in the direction of $\mathbf{a}$ .
Maximum Directional Derivative: * Occurs when the direction is exactly the same as the gradient vector. * Formula: $|\nabla f|$ .
Detailed Example: * Function: $f = xy + yz + zx$ at point $(1, 2, 0)$ in the direction of vector $\mathbf{a} = \mathbf{i} + 2\mathbf{j} + 2\mathbf{k}$ . * $∇f = (y + z)\mathbf{i} + (x + z)\mathbf{j} + (x + y)\mathbf{k}$ . * At $(1, 2, 0)$ , $\u2207f = 2\mathbf{i} + 1\mathbf{j} + 3\mathbf{k}$ . * Unit vector $\mathbf{\hat{a}} = \frac{\mathbf{i} + 2\mathbf{j} + 2\mathbf{k}}{\sqrt{1^2 + 2^2 + 2^2}} = \frac{\mathbf{i} + 2\mathbf{j} + 2\mathbf{k}}{3}$ . * D.D. calculation: $(2\mathbf{i} + \mathbf{j} + 3\mathbf{k}) \cdot \frac{(\mathbf{i} + 2\mathbf{j} + 2\mathbf{k})}{3} = \frac{(2 \times 1) + (1 \times 2) + (3 \times 2)}{3} = \frac{10}{3}$ .

Normal Vector (\mathbf{n}): * The gradient of a surface function at a specific point results in the normal vector to that surface: $\mathbf{n} = \nabla f$ .
Unit Normal Vector (\mathbf{\hat{n}}): * $\mathbf{\hat{n}} = \frac{\nabla f}{|\nabla f|}$ .
Angle Between Two Surfaces: * Defined by the angle between their respective normal vectors at the point of intersection. * Formula: $\cos(\theta) = \frac{\nabla f \cdot \nabla g}{|\nabla f| |\nabla g|}$ . * Applied Example: * Surface 1: $f = x^2 + y^2 + z^2 - 9 = 0$ . * Surface 2: $g = x^2 + y^2 - z - 3 = 0$ . * Point: $(2, -1, 2)$ . * $\nabla f = 2x\mathbf{i} + 2y\mathbf{j} + 2z\mathbf{k} \rightarrow 4\mathbf{i} - 2\mathbf{j} + 4\mathbf{k}$ . * $\nabla g = 2x\mathbf{i} + 2y\mathbf{j} - \mathbf{k} \rightarrow 4\mathbf{i} - 2\mathbf{j} - \mathbf{k}$ . * $\nabla f \cdot \nabla g = (4 \times 4) + (-2 \times -2) + (4 \times -1) = 16 + 4 - 4 = 16$ . * $|\nabla f| = \sqrt{16 + 4 + 16} = 6$ . * $|\nabla g| = \sqrt{16 + 4 + 1} = \sqrt{21}$ . * Result: $\cos(\theta) = \frac{16}{6\sqrt{21}} = \frac{8}{3\sqrt{21}}$ .

Divergence (\text{div } \mathbf{F}): * Represents the scalar output obtained by taking the dot product of the del operator and a vector point function. * Formula: $\text{div } \mathbf{F} = \nabla \cdot \mathbf{F} = \frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y} + \frac{\partial F_3}{\partial z}$ .
Solenoidal Vector: * A vector field $\mathbf{F}$ is called solenoidal if its divergence is zero ( $\nabla \cdot \mathbf{F} = 0$ ).
Example: * $\mathbf{F} = xy^2\mathbf{i} + 2x^2yz\mathbf{j} - 3yz^2\mathbf{k}$ . * $\text{div } \mathbf{F} = \frac{\partial}{\partial x}(xy^2) + \frac{\partial}{\partial y}(2x^2yz) + \frac{\partial}{\partial z}(-3yz^2)$ . * $\text{div } \mathbf{F} = y^2 + 2x^2z - 6yz$ . * At point $(1, 1, 1)$ : $1 + 2(1) - 6(1) = -3$ .

Curl (\text{curl } \mathbf{F}): * Represents the vector output obtained by taking the cross product of the del operator and a vector point function. * Formula: $\text{curl } \mathbf{F} = \nabla \times \mathbf{F} = \text{det} \begin{pmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ \partial/\partial x & \partial/\partial y & \partial/\partial z \ F_1 & F_2 & F_3 \end{pmatrix}$ .
Irrotational Vector: * A vector field $\mathbf{F}$ is called irrotational if its curl is zero ( $\nabla \times \mathbf{F} = \mathbf{0}$ ).
Example (continued from Divergence): * Function: $\mathbf{F} = xy^2\mathbf{i} + 2x^2yz\mathbf{j} - 3yz^2\mathbf{k}$ . * Partial calculation for $\mathbf{i}$ component: $\frac{\partial}{\partial y}(-3yz^2) - \frac{\partial}{\partial z}(2x^2yz) = -3z^2 - 2x^2y$ . * After full expansion and point evaluation at $(1, 1, 1)$ , the result for the curl is approximately $5\mathbf{i} + 0\mathbf{j} + 2\mathbf{k}$ .

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