Scalar Fields

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

Vector Fields

If to each point $\mathrm{P}(\vec{r})$ of a region $R$ in space there is a unique vector denoted by $\overrightarrow{\mathrm{F}}(\vec{r})$ then $\overrightarrow{\mathrm{F}}$ 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 $\lim _{\Delta t \rightarrow 0} \dfrac{\vec{f}(t+\Delta t)-\vec{f}(t)}{\Delta t}$ exists. Then it is denoted by $\dfrac{d \vec{f}}{d t}$ or $\vec{f}^{\prime}$ 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 $\lim _{\Delta \rightarrow 0} \dfrac{\Delta \vec{r}}{\Delta t}$ exists, it is denoted by $\dfrac{d \vec{r}}{d t}$ and $\dfrac{d \vec{r}}{d t}$ is in the direction of the tangent at $P$ to the curve.

If $\dfrac{d \vec{r}}{d t} \neq 0,$ then $\dfrac{d \vec{r}}{d t}$ or $\vec{r}^{\prime}(t)$ 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 $\vec{r}^{\prime}(t)$ 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 $\phi$ be a continuous scalar point function defined in a region $R$ in space. Then the set of all points satisfying the equation $\phi(x, y, z)=C,$ where $C$ is a constant, determines a surface which is called a level surface of $\phi$.

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 $\vec{\mathrm{A}}=(y z+2 x) \hat{i}+x z \hat{j}+(x y+2 z) \hat{k}$.

[2008, 15M]

2) Evaluate $\iint_{s} \vec{F} \cdot \hat{n} dS$, where $\vec{\mathrm{F}}=4 x \hat{i} -2y^2 \hat{j}+ z^2 \hat{k}$, 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 $\overline{r}=\sin t \hat{i}+\cos 2 t \hat{j}+\left(t^{2}+2 t\right) \hat{k}$. Find the components of acceleration $\overline{a}$ in the direction parallel to the velocity vector $\overline{v}$ and perpendicular to the plane of $\overline{r}$ and $\overline{v}$ at time $t=0$.

[2017, 10M]

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

[2011, 20M]

< Previous Next >