1. Find the directional derivative of the function φ = x y² + y z³ at the point (2, -1, 1) in the direction of the normal to the surface x log z - y² + 4 = 0, at ( -1, 2, 1 ). 2. F... 1. Find the directional derivative of the function φ = x y² + y z³ at the point (2, -1, 1) in the direction of the normal to the surface x log z - y² + 4 = 0, at ( -1, 2, 1 ). 2. Find div F, curl F where F = e^(x y² i + y z² j + z x² k) at the point (1, 2, 3). 3. Prove that (i) div (a x b) = b . curl a - a . curl b and (ii) curl(u a) = u . curl a + grad u x a. 4. Show that ∇rⁿ = n rⁿ⁻² r̂ and hence evaluate ∇¹/r where r̂ = x i + y j + z k. 5. Show that the vector field F = r̅/r³ is irrotational as well as solenoidal. Find the vector potential. 6. Prove that div (rⁿ r̂) = (n + 3) rⁿ further that rⁿ r̂ is solenoidal only if n = -3. 7. Using Green's theorem, find the area of the region in the first quadrant bounded by the curve y = x, y = 1/x, y = x/4. 8. Using Green's Divergence theorem, evaluate ∬ S[e^x dy dx - ye^x dz dx + 3z dx dy] where S is the surface of the cylinder x² + y² = c², 0 ≤ z ≤ h. Read more

Steps to Solve

1. Directional Derivative Calculation

To find the directional derivative of the function $\phi = x y^2 + y z^3$ at the point $(2, -1, 1)$ in the direction of the normal to the surface $x \log z - y^2 + 4 = 0$ at the point $(1, -2, 1)$, we first need to compute the gradient of $\phi$:

$$ \nabla \phi = \left( \frac{\partial \phi}{\partial x}, \frac{\partial \phi}{\partial y}, \frac{\partial \phi}{\partial z} \right) $$

Calculating the partial derivatives, we have:

$$ \frac{\partial \phi}{\partial x} = y^2, \quad \frac{\partial \phi}{\partial y} = 2xy + 3yz^2, \quad \frac{\partial \phi}{\partial z} = 3y^2z^2 $$

Evaluate at $(2, -1, 1)$ to find $\nabla \phi$.

2. Normal Vector Calculation

Next, we find the normal vector to the surface defined by $x \log z - y^2 + 4 = 0$. We compute the gradient of the surface function:

$$ \nabla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right) $$

where $f = x \log z - y^2 + 4$. Calculate the partial derivatives and evaluate at the point $(1, -2, 1)$.

3. Compute the Directional Derivative

Now we can find the directional derivative of $\phi$ in the direction of the normal vector:

$$ D_{\mathbf{n}} \phi = \nabla \phi \cdot \mathbf{n} $$

Use the dot product of $\nabla \phi$ and the normal vector $\mathbf{n}$ obtained from the previous step.

4. Divergence and Curl Calculation

For Q2, calculate the divergence and curl of the vector field $\mathbf{F} = e^{x} (x^2 y^2 \hat{i} + y^2 z^2 \hat{j} + z^2 x^2 \hat{k})$.

Divergence is given by:

$$ \text{div} \mathbf{F} = \frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y} + \frac{\partial F_3}{\partial z} $$

Curl is given by:

$$ \text{curl} \mathbf{F} = \nabla \times \mathbf{F} $$

5. Prove Div and Curl Identities

In Q3, verify the identity for $\text{div}(\mathbf{a} \times \mathbf{b}) = \mathbf{b} \cdot \nabla \times \mathbf{a} - \mathbf{a} \cdot \nabla \times \mathbf{b}$. Calculate each term appropriately to show the equivalence.

6. Identity Check with Laplacian

In Q4, show that

$$ \nabla r^n = n r^{n-2} \mathbf{r} $$

where $\mathbf{r} = xi + yj + zk$.

7. Irrational and Solenoidal Check

In Q5, determine if $\mathbf{F} = \frac{\mathbf{r}}{r^3}$ is both irrotational and solenoidal by examining $\text{div} \mathbf{F} = 0$ and $\text{curl} \mathbf{F} = 0$.

8. Divergence Proof for $r^n$

In Q6, prove that for the expression involving $\text{div}(r^n \mathbf{r}) = (n + 3) r^{n-2}$. Analyze the terms derived from the calculus of divergence.

9. Area Calculation with Green’s Theorem

In Q7, apply Green's theorem to calculate the area bounded by $y = x$ and $y = \frac{1}{x}$ in the first quadrant.

10. Surface Integral Evaluation

Lastly, in Q8, evaluate the triple integral involving the cylinder $x^2 + y^2 = c^2$, where $0 \leq z \leq h$, using cylindrical coordinates.