SageMath MCP Server

Vector Calculus Operation

vector_calculus_operation

Idempotent

Compute gradient, divergence, curl, or Laplacian of scalar or vector fields using provided expressions and variables, returning symbolic results.

Instructions

Vector calculus operations: gradient, divergence, curl, laplacian

Input Schema

TableJSON Schema

Name	Required	Description	Default
`session`	No	Workspace to use, as a name or a portable handle. Workspaces have independent variables. A name is scoped to this MCP session; a handle returned by start_sage_session (workspace_token) reaches the same workspace across reconnects and is a bearer credential -- keep it secret. Omit for 'default'.	default
`operation`	Yes	One of: gradient, divergence, curl, laplacian
`variables`	No	Variable names (e.g. ['x', 'y', 'z'])
`expression`	Yes	Scalar field (string) for gradient/laplacian, or vector field components (list) for divergence/curl

Output Schema

TableJSON Schema

Name	Required	Description	Default
No arguments

Schema Changelog

Changes observed during successful MCP inspections.

Changed1 schema field changedv0.7.0Sep 12, 2026

changedInput schema / properties / session / description

Previous value: -"Named workspace to use. Workspaces have independent variables; omit for 'default'."New value: +"Workspace to use, as a name or a portable handle. Workspaces have independent variables. A name is scoped to this MCP session; a handle returned by start_sage_session (workspace_token) reaches the same workspace across reconnects and is a bearer credential -- keep it secret. Omit for 'default'."

Changed1 schema field changedv0.5.0Aug 14, 2026

addedInput schema / properties / session

Added value: +{
+  "default": "default",
+  "description": "Named workspace to use. Workspaces have independent variables; omit for 'default'.",
+  "type": "string"
+}

First observedv0.3.1Jul 31, 2026

TDQS

C2.8/5.0

Behavior2/5

Does the description disclose side effects, auth requirements, rate limits, or destructive behavior?

Annotations already indicate idempotentHint=true and destructiveHint=false, but the description adds no further behavioral context. It doesn't mention session requirements, side effects, or any operational details beyond the operation names. With readOnlyHint=false, it's unclear if there are side effects, and the description offers no clarification.

Agents need to know what a tool does to the world before calling it. Descriptions should go beyond structured annotations to explain consequences.

Conciseness3/5

Is the description appropriately sized, front-loaded, and free of redundancy?

The description is a single concise sentence with no waste, but it lacks structure and detail. It front-loads the operations list but doesn't provide any elaborative sections or guidance. It is appropriately brief but could benefit from a bit more explanatory structure without becoming verbose.

Shorter descriptions cost fewer tokens and are easier for agents to parse. Every sentence should earn its place.

Completeness2/5

Given the tool's complexity, does the description cover enough for an agent to succeed on first attempt?

For a tool with multiple operations and complex input types (scalar vs vector fields), the description is minimal. While the schema clarifies parameter formats, the description doesn't explain the distinction between scalar and vector inputs, or how operations map to them. It also doesn't cover the session parameter's purpose. Given the output schema exists, return values aren't needed, but usage context is lacking.

Complex tools with many parameters or behaviors need more documentation. Simple tools need less. This dimension scales expectations accordingly.

Parameters3/5

Does the description clarify parameter syntax, constraints, interactions, or defaults beyond what the schema provides?

The schema provides comprehensive descriptions for all parameters (operation, expression, variables, session), achieving 100% coverage. The description adds no additional semantic meaning beyond what the schema already documents, so the baseline score of 3 is appropriate.

Input schemas describe structure but not intent. Descriptions should explain non-obvious parameter relationships and valid value ranges.

Purpose4/5

Does the description clearly state what the tool does and how it differs from similar tools?

The description clearly identifies the tool as performing vector calculus operations and lists the specific operations (gradient, divergence, curl, laplacian). This distinguishes it from siblings like differentiate_expression or integrate_expression, though it doesn't explicitly state it computes or evaluates these operations.

Agents choose between tools based on descriptions. A clear purpose with a specific verb and resource helps agents select the right tool.

Usage Guidelines2/5

Does the description explain when to use this tool, when not to, or what alternatives exist?

No guidance is provided on when to use this tool versus alternatives. It doesn't mention any exclusions or context for selecting vector calculus operations, nor does it reference sibling tools. The description simply states what it does without indicating preferred use cases.

Agents often have multiple tools that could apply. Explicit usage guidance like "use X instead of Y when Z" prevents misuse.