Scalar Fields

If to each point $P (\vec{r})$ (the point $P$ with position vector $\vec{r}$ ) of a region $R$ in space there is a unique scalar or real number denoted by $ϕ (\vec{r}),$ then $ϕ$ is called a scalar point function in $R$ . The region $R$ is called a scalar field.

Vector Fields

If to each point $P (\vec{r})$ of a region $R$ in space there is a unique vector denoted by $\vec{F} (\vec{r})$ then $\vec{F}$ is called a vector point function in $R .$ The region $R$ is called a vector field.

Derivative of a Vector Function

A vector function $\vec{f} (t)$ is said to be differentiable at a point $t,$ if $lim_{Δ t \to 0} \frac{\vec{f} (t + Δ t) - \vec{f} (t)}{Δ t}$ exists. Then it is denoted by $\frac{d \vec{f}}{d t}$ or ${\vec{f}}^{'}$ and is called the derivative of the vector function $\vec{f}$ at $t$ .

Geometrical Meaning of Derivative

Let $\vec{r} (t)$ be the position vector of a point $P$ with respect to the origin $O$ . As $t$ varies continuously over a time interval $P$ traces the curve $C$ . Thus, the vector function $\vec{r} (t)$ represents a curve $C$ in space.

If $lim_{Δ \to 0} \frac{Δ \vec{r}}{Δ t}$ exists, it is denoted by $\frac{d \vec{r}}{d t}$ and $\frac{d \vec{r}}{d t}$ is in the direction of the tangent at $P$ to the curve.

If $\frac{d \vec{r}}{d t} \neq 0,$ then $\frac{d \vec{r}}{d t}$ or ${\vec{r}}^{'} (t)$ is called a tangent vector to the curve $C$ at $P$ .

The unit tangent vector at

P

is $$=\dfrac{\vec{r}^{\prime}(t)}{\left

\vec{r}^{\prime}(t)\right

}=\hat{u}(t)$$

Both ${\vec{r}}^{'} (t)$ and $\hat{u} (t)$ are in the direction of increasing $t$ . Hence, their sense depends on the orientation of the curve $C$ .

Level Surface

Let $ϕ$ be a continuous scalar point function defined in a region $R$ in space. Then the set of all points satisfying the equation $ϕ (x, y, z) = C,$ where $C$ is a constant, determines a surface which is called a level surface of $ϕ$ .

PYQs

Vector Fields

1) Evaluate $\int_{c} \vec{A} \cdot d \vec{r}$ along the curve $x^{2} + y^{2} = 1$ , $z = 1$ from $(0, 1, 1)$ to $(1, 0, 1)$ if $\vec{A} = (y z + 2 x) \hat{i} + x z \hat{j} + (x y + 2 z) \hat{k}$ .

[2008, 15M]

2) Evaluate $\iint_{s} \vec{F} \cdot \hat{n} d S$ , where $\vec{F} = 4 x \hat{i} - 2 y^{2} \hat{j} + z^{2} \hat{k}$ , and $S$ is the surface of the cylinder bounded by $x^{2} + y^{2} = 4$ , $z = 0$ and $z = 3$ .

[2008, 15M]

Differentiation of a Vector Field of a Scalar Variable

1) The position vector of a moving point at time $t$ is $\bar{r} = \sin t \hat{i} + \cos 2 t \hat{j} + (t^{2} + 2 t) \hat{k}$ . Find the components of acceleration $\bar{a}$ in the direction parallel to the velocity vector $\bar{v}$ and perpendicular to the plane of $\bar{r}$ and $\bar{v}$ at time $t = 0$ .

[2017, 10M]

2) For two vectors $\vec{a}$ and $\vec{b}$ given respectively by $\vec{a} = 5 t^{2} \hat{i} + \hat{t} \hat{ȷ} - t^{3} \hat{k}$ and $\vec{b} = \sin 5 t \hat{i} - \cos t \hat{j}$ , determine:
i) $\frac{d}{d t} (\vec{a} \cdot \vec{b})$
ii) $\frac{d}{d t} (\vec{a} \times \vec{b})$

[2011, 20M]

< Previous Next >